Computational Aspects of Maass waveforms

ACTA UNIVERSITATIS UPSALIENSIS
Uppsala Dissertations in Mathematics
39

Fredrik Strömberg
Computational Aspects of Maass
Waveforms

Dissertation at Uppsala University to be publicly examined in Room 2347, Polacksbacken,
Tuesday, March 15, 2005 at 13:15 for the Degree of Doctor of Philosophy. The examination
will be conducted in English.
Abstract
Strömberg, F. 2005. Computational Aspects of Maass Waveforms. Uppsala Dissertations in
Mathematics 39. x, 176 pp. Uppsala. ISBN 91-506-1794-X
The topic of this thesis is computation of Maass waveforms, and we consider a number
of different cases: Congruence subgroups of PSL(2,Z) and Dirichlet characters (chapter 1);
congruence subgroups and general multiplier systems and real weight (chapter 2); and noncongruence
subgroups of the Modular group (chapter 3). In each case we first discuss the necessary
theoretical background. We then outline the algorithm and display some of the results obtained
by it.
Keywords: Maass waveform, Congruence group, Noncongruence group, Multiplier system,
Theta multiplier system, Eta multiplier system, weight, Hecke operators, Involution, Dirichlet
character, Spectral theory, Computation, Whittaker function
Fredrik Strömberg, Department of Mathematics, Uppsala University, Lägerhyddsvägen 2 ,Box
480, SE-75106 Uppsala, Sweden
c○ Fredrik Strömberg 2005
ISBN 91-506-1794-X
ISSN 1401-2049
urn:nbn:se:uu:diva-4778 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4778)

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Maass Waveforms on Hecke Congruence subgroups with Dirichlet
characters
(Computational Aspects) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 General Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 A Brief Introduction to Fuchsian Groups . . . . . . . . . . . . . 12
1.1.2 Hecke Congruence Groups . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.3 Introduction to Dirichlet Characters . . . . . . . . . . . . . . . . 16
1.1.4 A Brief Introduction to Maass Waveforms . . . . . . . . . . . . 18
1.2 Some Structural Theory of M(Γ0(N), χ,λ) . . . . . . . . . . . . . . . 20
1.2.1 The Conjugation Operator . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.2 The Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.3 Involutions and Normalizers . . . . . . . . . . . . . . . . . . . . . . 21
1.2.4 The Reflection Operator . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.5 Complete Symmetrization . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.6 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.7 Oldforms and Newforms . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2.8 The Cusp Normalizing Maps as Normalizers of (Γ0(N), χ) 32
1.3 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3.2 Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3.3 Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.4 Remarks on the Performance of the Algorithm . . . . . . . . . 44
1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.4.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4.2 Lowest Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4.3 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 Computation of Maass Waveforms with Non-trivial Multiplier Systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Multiplier Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.2 The η multiplier system . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.3 The θ Multiplier System . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii

2.2.4 Further properties of the multiplier systems . . . . . . . . . . . 61
2.3 Maass Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.1 Decomposition of the discrete spectrum . . . . . . . . . . . . . . 64
2.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4.1 Conjugation and reflection . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4.2 The involution ωN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4.3 The operator σ4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.4 Maass operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.5 Maass operators and the symmetry about k = 6 . . . . . . . . 72
2.4.6 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.4.7 Lifts at weight 1 and Fourier coefficients . . . . . . . . . . . . . 77
2.5 Oldforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5.1 Γ0(N) with N prime and η-multiplier . . . . . . . . . . . . . . . 81
2.6 The Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.6.1 Weight 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 Half integer weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7.1 The Shimura correspondence . . . . . . . . . . . . . . . . . . . . . . 83
2.8 Some Computational Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.9 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.9.1 Varying weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.9.2 Small weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.9.3 Lifts at weight 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.9.4 Half integer weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Computation of Maass waveforms on Non-congruence Subgroups of
the Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2 Subgroups of the Modular Group . . . . . . . . . . . . . . . . . . . . . . . 108
3.2.1 Permutations and subgroups of PSL(2,Z) . . . . . . . . . . . . 109
3.2.2 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.4.1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.4.2 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.3 Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4 An Algorithm for Whittaker’s W-function . . . . . . . . . . . . . . . . . . . 141
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2 Presentation of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3 The path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.4 The integrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.5 Evaluation of the Integrand . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.5.1 The power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.5.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
viii

4.5.3 Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.5.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.6 Evaluation of the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.7 Remarks on the K-Bessel function . . . . . . . . . . . . . . . . . . . . . . 156
4.8 Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Swedish Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
ix

Introduction
This is a dissertation in experimental mathematics, and we are thus in a situation
where the “pure” mathematics takes advantage of numerical results to
build hypotheses.
A short description of what distinguishes experimental mathematics is that
by using theoretical reasoning you construct numerical models (algorithms)
which are then used to generate information. On the basis of the numerical
results one then formulates hypotheses and conjectures (for a more detailed
description see [16]).
The prospect is of course that later one will be able provide rigorous proofs
of these conjectures, but even in those situations that the present-day mathematical
machinery is not up to this, the numerical results can provide valuable
insight and inspire to further studies.
The three chapters of the dissertation are all designed in the same way; the
bulk is devoted to a presentation of the theory, which is then used to construct
the algorithms. After a brief description of how the algorithms are designed,
we present our experimental findings.
The problem we have studied is the construction of the mathematical objects
referred to as “Maass waveforms” (after Hans Maass (1911-1992)). Mathematical
applications of these wavefunctions are mainly in analytical number
theory. There are also connections to physics, foremost to mathematical models
of quantum chaos ([35, 14]), but also to cosmology ([8]).
On the Physical Background
Consider the motion of a free (quantum mechanical) particle with mass m0
on a surface S . According to quantum mechanics, this system is described
by a wave function, ψ, which tells us about the probability of encountering
the particle in a certain region of the surface. The propagation of this wave
function, for a particle with the energy E = ¯hv, is governed by the Schrödinger
equation:
i¯h ∂ψ
∂t
¯h2
= − ∆ψ = Eψ,
2m0
1

where ∆ is the so-called Laplace-Beltrami operator on the surface. (e.g. in the
plane: ∆ψ = ψ ′′
xx + ψ ′′
yy). We now assume that we can separate ψ as a product
of a time-dependent part, e−ivt , and a time-independent part, φ(P), P ∈ S . We
thus obtain an equation for the time independent part (the time-independent
Schrödinger equation, or the Laplace equation):
∆φ + λφ = 0, (⋆)
where we have set λ = 2m0E
¯h 2 (the eigenvalue), and since the probability is one
that the particle is somewhere on the surface we must have
|φ| 2 dP = 1.
S
If we have a charged particle and introduce a constant magnetic field, B, the
time independent part of the wave function will obtain a “twist” (spin) and φ
will now satisfy a modified Laplace equation:
∆Bφ + λφ = 0, (⋆⋆)
where ∆B is a modified Laplace-Beltrami operator, (in our case, it will read
∆B = ∆ − iyB ∂
∂x , with B proportional to the magnetic field strength).
Observe that the wave equation (⋆) is “universal”, in the sense that it describes
many different kinds of wave motion. For example, if S describes a
drum, then (⋆) will describe the appearance of a standing wave on the drum
skin, with a frequency proportional to √ λ. Bass notes, thus correspond to
small λ and higher notes to larger λ.
In this thesis we consider the equations (⋆) and (⋆⋆) for certain special surfaces,
S , namely those with constant negative curvature (we say that the surface
is hyperbolic), finite area and a finite (positive) number of “punctures”
or “cusps” (i.e. we have removed a finite number of points from the surface
and pulled them away towards infinity). Furthermore, the surfaces have to
be special in an arithmetical way, which in this case means that they correspond
to subgroups of the modular group. We also allow our surfaces to have
“corners” (marked points). The solutions to (⋆) and (⋆⋆) on such surfaces are
called Maass waveforms.
One reason that surfaces of negative curvature are of interest in this context
is that the classical free-particle motion on these surfaces is known to be
(strongly) chaotic.
By studying the solutions of (⋆) on such a surface we might thus gain some
insight about “traces” of this classical chaos left behind in the quantized system.
This type of investigations is part of the research area quantum chaos.
The wavefunctions are not chaotic in themselves, and the classical chaos is
instead visible in e.g. the distribution of the energy levels (eigenvalues) and
2

in the appearance of the wavefunctions φ which corresponds to high energies.
In Figure 2 you can see a sample of how a Maass waveform, ϕ with an
eigenvalue λ ≈ 10000 looks. (a) Shows the nodal lines (i.e. the points where
ϕ(z) = 0), and in (b) you can see the probability function associated to ϕ,
i.e. |ϕ(z)| 2 . The light regions in (b) shows the places in the plane where the
quantum mechanical particle likes to hang around.
The main purpose with the work underlying this thesis was to obtain stable
algorithms to compute eigenvalues, λ, and corresponding wave functions, φ,
to the above mentioned problems.
A brief Introduction to Hyperbolic Geometry
To visualize negative curvature (hyperbolic geometry), as a mathematician
one usually works in the upper half-plane model, i.e.
H = {x + iy ∈ C|y > 0},
where we introduce a concept of distance and area with the help of the arc
element
ds = |dz|
y ,
and the area element
dµ = dxdy
.
y2 The shortest path between two points z1,z2 ∈ H is obtained by following
the geodesic which passes through these points. In the usual (flat) case the
geodesic is the unique straight line which passes through these points. In the
hyperbolic upper half plane, the geodesics are the semi-circles and straight
lines which are perpendicular to the x-axis. See Figure 1 for an illustration of
Figure 1: The upper half-plane
H
γ3
y
z4
z3
γ1
γ2
z2
z1
(a) Geodesics and a horocycle
x
S
F1 F4 F7
E
e2
e3 e ′ 3
F6
F3
F5
F2
(b) A fundamental domain
3

the geodesics γ1 and γ2 which connect z1 with z2 and z3 with z4 respectively.
In the figure we also give an example of a horocycle, γ3.
A consequence of this (odd) way to measure distances in the upper halfplane
is for example that you can never get all the way down to the real line,
y = 0 (i.e. the distance from any point in H to the real line is infinite).
A more physical way to think about the hyperbolic space is the following
(after Poincaré [82]):
"Visualize yourself in a finite spherical universe, bounded by a “shell”. Assume
that the temperature in this universe is proportional to the distance from
the shell. That is, it is hottest in the center, and the absolute zero temperature is
attained at the shell. Now, we also make the assumption that the speed of movement
is proportional to the temperature (this is what one expect physically, at
the molecular/atomic level).
If we now decide to move to the end of the universe (i.e. the shell) it is easy
to realize that we will never get there (in finite time). What we will experience
is that the closer we get to the shell, the lower the temperature gets and the
slower we move.
Mathematically, under these assumptions, we can can compute our distance
from the center at the time t, r(t) = R(1 − e −ct ), where R is the radius of the
shell and c is a positive constant."
An isometry in a space (where we have a concept of distances) is a mapping
which preserves distances and areas. In the usual “flat” case, the translations,
rotations and reflections are examples of isometries. Reflection is an example
of an orientation-reversing isometry, while translations and rotations are examples
of orientation-preserving isometries. In the upper half-plane we know
that any orientation preserving isometry can be written as a Möbius transfor-
mation:
T : z ↦→
az + b
, where a,b,c,d ∈ R, ad − bc = 1.
cz + d
Such a transformation is then associated to a pair of matrices:
±
a
c
b
d
,
and the group of all such isometries are usually denoted by PSL(2,R) =the
group of all matrices
a b
c d , with a,b,c,d ∈ R, ad − bc = 1, and where we
identify the matrices
a b −a −b
c d and −c −d .
One knows that all hyperbolic surfaces can be described with the help of
discrete subgroups of PSL(2,R), i.e. we can write our surface S as S =
Γ\H , where Γ is a Fuchsian group (a discrete group of Möbius transformations).
Those surfaces, S , we are interested in have “horns” (points at ∞) and
4

they can also have corners. In the corresponding group, Γ, the horns show
themselves as fixed-points to parabolic transformations, i.e. those that correspond
to matrices with the absolute value of the trace = 2 (remember that the
trace of a matrix
a b
c d is defined as a + d). The corners, on the other hand are
fixed-points of elliptic (|trace|< 2). Therefore, one usually says that the corresponding
points are parabolic or elliptic. (The parabolic fixed-points are also
called cusps). In PSL(2,R), in addition to elliptic and parabolic transformations
(which geometrically correspond to rotations and translations) also hyperbolic
transformations (|trace|> 2). While the first types have unique fixedpoints,
in general the hyperbolic maps have two fixed points in R ∪ {∞} (and
geometrically, they describe a movement along the geodesic which connects
these points).
To illustrate the surface S = Γ\H it is customary to use a fundamental
domain, which in our case can be taken as a connected domain F ⊂ H with
the following two properties: we can not map any point inside F to another
point inside F with a map in Γ, and secondly, the Γ-images F must tessellate
H (i.e. ∪γ∈Γγ(F ) = H). An example of a fundamental domain is the
standard fundamental domain for the modular group, PSL(2,Z) (all matrices
in PSL(2,Z) with integer entries), F1 = {z = x + iy ∈ H||z| ≥ 1, |x| ≤ 1
2 }.
In Figure 1(b) we can see F1 together with 6 “copies”. Observe that all images,
F j, have the same hyperbolic area. If we identify the sides of F1 using
the maps S : z ↦→ z + 1 and E : z ↦→ −1 we get (topologically) a sphere with
z
one cusp (i∞) and two corners (e2 and e3 which is identified with e ′ 3
). One can
show that the side pairing transformations E and S generates PSL(2,Z) and,
in fact, in all cases we consider here, we can always choose the fundamental
domain in such a way that the side pairing maps generate the group.
Summary of the Chapters in the Thesis
In chapter one we study the problem which in the upper half-plane model can
be written as:
y 2 φ ′′
xx(z) + φ ′′
yy(z) + λφ(z) = 0, z = x + iy ∈ H , (1)
φ (T z) = χ(T )φ(z),
T ∈ Γ, z ∈ H , (2)
|φ| 2 dµ = 1, (3)
Γ\H
where χ : Γ → S 1 as an even Dirichlet character (i.e. χ(AB) = χ(A)χ(B) and
χ(−1) = 1), and Γ is a Hecke-congruence subgroup of the modular group. For
a given positive integer N, we define the Hecke-congruence group, Γ0(N), in
5

the following way:
Γ0(N) =
a
c
b
d
|a,b,c,d ∈ Z, ad − bc = 1, c ≡ 0 mod N .
We will now briefly outline the algorithm which we use to find solutions of this
problem. Given an eigenvalue, λ = 1
4 + R2 , we know that the corresponding
φ can be written as a Fourier series:
φ(x + iy) = ∑ c(n)
n=0
√ yKiR(2π|n|y)e 2πinx ,
where KiR(y) is the K-Bessel function. Due to the rapid decay of KiR(y) we
can approximate φ (up to any given ε > 0) with a truncated Fourier series,
φ(x + iy) ≈ ˆφ(x + iy) = ∑
0 0 is chosen wisely), thus providing us with a set of linear equations for
the coefficients c(n):
c(n) √ Y KiR(2π|n|Y ) = 1
2Q
Q
∑
m=1−Q
ˆφ(zm)e(−nxm) ≈ 1
2Q
Q
∑
m=1−Q
φ(zm)e(−nxm),
and by using the implicit automorphy (2) we can replace the point zm with a
point, z ∗ m = x ∗ m + iy ∗ m in the fundamental domain of the group. Doing this, we
get a non-trivial, well-conditioned, linear system of equations
c(n) √ Y KiR(2π|n|Y ) = ∑ Vnkc(k),
|k|≤M0
(4)
which we can solve to obtain the Fourier coefficients c(n).
If we do this for an arbitrary value of R, the resulting function φ will obviously
not be invariant under Γ. But if 1
4 + R2 is close to a true eigenvalue
the computed Fourier coefficients will be close to the coefficients of a true
eigenfunction. In particular, for a “true” R, the solution vectors should be independent
of the value of Y (provided it is kept small enough that the system
does not become trivial).
We use this fact to locate eigenvalues as follows; using two different values
of Y in parallel, we solve (4) over and over again, for R running through a
specified interval, all the while we try to minimize the functional
6
h(R) = c(2) − c ′ (2) + c(3) − c ′ (3) + c(4) − c ′ (4) ,

where c(n) and c ′ (n) are the solutions of (4) corresponding to the different
Y ’s. When we find a minimizing R-value, we go through with further (more
technical, e.g. using Hecke-relations) tests to see if we indeed seem to be close
to a true eigenvalue.
As soon as we have an approximate eigenvalue and a corresponding set of
Fourier coefficients we can use another algorithm (phase 2) to easily compute
many more coefficients.
In chapter 2 we study a similar problem, where we instead of (1) consider
the equation
y 2 φ ′′
xx(z) + φ ′′
yy(z) − iymφ ′ x + λφ(z) = 0,z = x + iy ∈ H , (1’)
where the weight, m, is a real number, and where the condition (2) is replaced
by
φ(T z) = v(T ) jT (z;m)φ(z), (2’)
where v : Γ → C is a so-called multiplier system and
jT (z;m) = exp(2πimArg(cz + d)),
for T =
a b
c d and m ∈ R.
Physically, we can interpret the equation (1’) as if we now have a charged
(quantum mechanical) particle moving in a constant magnetic field on the surface
Γ\H. The magnetic field strength is proportional to m, and the multiplier
system can be used to describe the magnetic flux through different parts of the
surface (the horns, handles etc.) Cf. [2, 1, 11]. (For those that are knowledgeable
in the physics here, (1’) corresponds to the choice of the Landau gauge,
A = c m
y dx for some suitable constant c.)
It turns out that the method introduced in chapter one is still applicable after
a few (minor) changes. The (computationally) hardest problem here is that the
Fourier series now contain the Whittaker W-function instead of the K-Bessel
function. The W-function is more difficult to compute, and we were forced
to develop a new algorithm for that purpose. The algorithm is presented in
Chapter 4.
In chapter 3 we investigate the same problem as in chapter 1, but here
we also consider subgroups Γ ⊆ PSL(2,Z) which are not congruence subgroups.
The main difference between congruence subgroups and general noncongruence
subgroups is that one doesn’t have as many “symmetries” available
in the noncongruence case. This implies for example that one doesn’t
have much a priori knowledge of neither the spectra nor the eigenfunctions.
And one can therefore pose questions which are more or less answered in the
congruence case: Are there any “new” Maass waveforms? If so, how many
are there? Do they show any symmetries? How are their Fourier coefficients
7

distributed?
We prove the existence of new Maass waveforms for groups Γ which have
certain specific properties, which on the surfaces Γ\H basically correspond to
different forms of mirror symmetries. (To systematically verify these properties
for large groups, computer aid is a necessity).
8

Figure 2: Plots of a Maass waveform, ϕ(z) on Γ0(5) with R = 100.0870...
2.1
Nodal lines for R=100.0870... on Γ 0 (5)
0.1
−1 1
(a) Nodal lines
(b) Density plot, i.e. |ϕ(z)| 2
9

1 Maass Waveforms on Hecke Congruence
subgroups with Dirichlet characters
(Computational Aspects)
Introduction
The main topic of this chapter is computational aspects of the theory of Maass
waveforms, i.e. square-integrable eigenfunctions of the Laplace- Beltrami operator,
on certain Riemann surfaces with constant negative curvature and finite
area. The surfaces under consideration correspond to quotients of the upper
half-plane by certain discrete groups of isometries, the so called Hecke congruence
subgroups.
It is known that such functions can also be regarded as wavefunctions corresponding
to a quantum-mechanical system describing a particle moving freely
on the surface. Since the classical counterpart of this motion is chaotic, the
study of these wavefunctions is closely related to the study of quantum chaos
on the surface. The presentation here, however, will be purely from a mathematical
viewpoint.
Today, our best knowledge of generic Maass waveforms comes from numerical
experiments, and these have previously been limited to the modular
group, PSL(2,Z), and certain triangle groups (cf. [43, 45, 42] and [98]). One
of the primary goals of this lecture is to give the necessary theoretical background
to generalize these numerical experiments to Hecke congruence subgroups
and non-trivial characters. We will describe algorithms that can be
used to locate eigenvalues and eigenfunctions, and we will also present some
of the results obtained by those algorithms.
There are four sections, first elementary notations and definitions, mostly
from the study of Fuchsian groups and hyperbolic geometry. Then more of the
theoretical background needed to understand the rich structure of the space of
Maass waveforms will be introduced. The third section deals with the computational
aspects, and the final section contains some numerical results.
11

1.1 General Definitions and Notation
1.1.1 A Brief Introduction to Fuchsian Groups
For a more thorough (but still elementary) introduction to this subject see for
example [52] or [31].
The following notation will be used; M2(K) is the ring of 2 × 2 matrices
over some base ring K, usually R or Z, GL(2,K) ⊆ M2(K) is the group of
invertible 2 × 2 matrices and SL(2,K) ⊆ M2(K) is the group of matrices with
determinant equal to 1. H denotes the Poincaré upper half-plane {z = x +
iy|y > 0} equipped with the hyperbolic metric and area measure
ds 2 = 1
y (dx2 + dy 2 ), dµ = 1
y
2 dxdy.
PSL(2,R) is the group of Möbius transformations with real coefficients,
az + b
PSL(2,R) = z →
cz + d a,b,c,d ∈ R, ad − bc = 1 .
It is clear that we can represent such mappings with matrices and there is an
isomorphism PSL(2,R) ≈ SL(2,R)/{±Id}, where Id is the identity element
in SL(2,R). We will use matrix and transformation notation interchangeably,
all groups discussed are subgroups of PSL(2,R) so they always contain −Id,
hence there should be no confusion when we use the same notation for matrix
groups as for transformation groups. Note that H, the hyperbolic metric and
the hyperbolic measure are invariant under PSL(2,R). The only subgroups of
PSL(2,R) that we are interested in are the discrete subgroups.
Definition 1.1.1. A Fuchsian group is a discrete subgroup of PSL(2,R).
The basic example of a Fuchsian group, for us, is the modular group,
PSL(2,Z) = SL(2,Z)/{±Id} ⊆ PSL(2,R).
This is just the subgroup of Möbius transformations in PSL(2,R) with integer
coefficients. This is also an example of what is called an arithmetic group
(cf. [52, ch. 5]) and is of interest for number theorists among others.
If Γ is a Fuchsian group it is a standard result that the space of Γ−orbits,
Γ\H = {Γz|z ∈ H}, can be given the analytical structure of a Riemann surface
with marked points (this is also called an orbifold) (cf. [65, II.F.], and for
more details [85]). On the other hand, a classical result, the Klein-Poincaré
uniformization theorem, asserts that any Riemann surface M with constant
negative curvature equal to −1 can be realized as Γ\H for some Fuchsian
group Γ (see [60] for more details on uniformization). We can visualize the
Riemann surface Γ\H via a fundamental domain F for Γ.
12

Definition 1.1.2. A closed set F ⊆ H is a fundamental domain for Γ if
a) ∪T ∈ΓT (F) = H, and
b) if F ◦ denotes the interior of F then T1(F ◦ ) ∩ T2(F ◦ ) = /0 if T1 =
T2 ∈ Γ.
Note that we use closed fundamental domains in line with for example [52],
so they will in general contain some equivalent boundary points. There are
two important examples of fundamental domains.
Definition 1.1.3. Let Γ be a Fuchsian group and let p ∈ H be a point that
is not fixed by any element in Γ. Let d(z,w) denote the hyperbolic distance
between z and w and define
DΓ(p) = {z ∈ H|d(z, p) ≤ d(z,T p), ∀T ∈ Γ}.
The set DΓ(p) is called a Dirichlet fundamental domain for Γ, with center p.
Let Γz = {T ∈ Γ|T z = z} denote the stabilizer in Γ of the point z ∈ H ∪R∪
{∞}. Another important type of fundamental domain, FΓ, the Ford fundamental
domain, can be described in terms of the exterior of isometric circles. Let
DT = z ∈ H |T ′ (z)| ≥ 1 ,∀T ∈ PSL(2,R). Then, if Γ∞ = {T ∈ Γ|T ∞ = ∞}
is non-empty and generated by S : z ↦→ z + 1, we define
FΓ =
T ∈Γ\Γ∞
DT ∩
z ∈ H |ℜz| ≤ 1
.
2
The boundary (in H ∪ R ∪ {∞}) of a fundamental domain F might contain
vertices that are fixed-points of elements of Γ; these points are called elliptic
or parabolic vertices respectively if the transformations fixing them are either
elliptic or parabolic. A parabolic vertex is usually referred to as a cusp, and is
viewed as either a point removed from the surface Γ\H, or a point at infinity.
1.1.2 Hecke Congruence Groups
Let N be any positive integer. We define the principal congruence subgroup
of level N, Γ(N) ⊆ PSL(2,Z), by
a b
Γ(N) =
∈ PSL(2,Z)
a b 1 0
c d
≡
mod N .
c d 0 1
This is a subgroup of finite index in PSL(2,Z), and any subgroup of PSL(2,Z)
containing some Γ(N) is called a congruence subgroup. The Hecke congru-
13

ence subgroup, Γ0(N), is defined by
a b
Γ0(N) =
∈ PSL(2,Z)
c d
c ≡ 0 mod N .
It is obvious that Γ(N) ⊆ Γ0(N) so Γ0(N) is a congruence subgroup.
The standard fundamental domain for PSL(2,Z) = Γ0(1) is the set
F1 = z ∈ H
1
|z| ≥ 1, |ℜz| ≤ ,
2
which is also a Ford fundamental domain. Suppose from now on that we have
fixed a set of right coset representatives {Vk} vN
k=1 for Γ0(N) in Γ0(1). (See
pp. 43–43 on how to construct such maps Vk.) One knows that
−1
vN = [Γ0(1) : Γ0(N)] = N∏ 1 + p
p|N
.
We can now fix a fundamental domain for Γ0(N) corresponding to these coset
representatives (cf. [52, thm. 3.1.2])
FN = ∪ vN
k=1 Vk(F1). (1.1)
Note that FN need not be normal in the sense of [39], but it will be bounded
by finitely many geodesics and it also has some other nice properties as we
will see (Theorem 1.1). Denote by κ0 the total number of parabolic vertices
of FN, and by κ the number of inequivalent cusps of FN (note κ ≤ κ0). It is
known that
κ = ∑ φ d,
d|N,d>0
N
,
d
where φ is Eulers totient function,
φ(n) = # j ∈ {1,2,3,...,n} ( j,n) = 1
(see [51, 89]). Let v1,...,vκ0 be the set of parabolic vertices of FN and
p1,..., pκ a set of inequivalent cusps. Without loss of generality we set p1 =
∞.
Definition 1.1.4. A cusp normalizing map associated to a cusp p j, is a map
σ j ∈ PSL(2,R) satisfying
i) σ j(∞) = p j, and
ii) σ jSσ −1
j = S j,
where Sz = z + 1 is a generator of Γ∞, and S j is a generator of Γp j , the (cyclic
14

infinite) stabilizer group of the cusp p j. Then σ j is uniquely determined up to
a translation.
For the groups Γ0(N) we will always choose the cusp normalizing maps σ j
of the form σ j = A jρ j, where A j ∈ SL(2,Z) maps ∞ to p j and ρ j is a scaling
by h j, the so-called width of the cusp p j (cf. [51, pp. 36-37]). It is important
to note that h j|N. For the cusp p1 = ∞ we will always take σ1 as the identity.
We will now provide an important fact about the fundamental domain for
Γ0(N) that will be of use to us later. First, for each parabolic vertex vℓ of
FN we choose a map Uℓ ∈ Γ0(N) which maps vℓ to its cusp representative in
{p1,..., pκ}; say Uℓ(νℓ) = p j(ℓ). If vℓ is already a cusp representative we take
Uℓ = Id. This will give us a finite collection of maps U = {Uℓ}1≤ℓ≤κ0 . Now,
to any point w ∈ FN we associate the “closest” (see Remark 1.1.1) parabolic
vertex, vw, and a corresponding map Uw = Uℓ(vw) ∈ U .
Remark 1.1.1. By the “closest” vertex to a point w ∈ FN we mean the vertex
vℓ with respect to which the point w has the greatest height, i.e. that vℓ for
which ℑ(σ −1
j(ℓ) Uℓw) is maximal. We use I(w) to denote the index of the cusp
representative corresponding to the vertex closest to w.
Define the height of w ∈ FN, h(w), by
h(w) = ℑ
σ −1
I(w) Uw (w)
and the minimal height of the fundamental domain FN, Y0, by
Y0 = Y0(N) = inf h(w) = inf ℑ σ
w∈FN w∈FN
−1
I(w) Uw
(w) . (1.2)
It is clear that Y0 depends only on the fundamental domain FN (i.e. the choice
of the coset representatives {Vk}) and not on the choice of the cusp representatives
{p j}. A standard compactness argument also implies that Y0 is strictly
positive and larger than some fixed quantity only depending on the fundamental
domain, and the following theorem actually gives us an explicit bound.
Theorem 1.1. For any positive integer N, the minimal height of FN satisfies
the following inequality
√
3
Y0(N) ≥
2N .
Proof. Let {Vk} be the fixed set of coset representatives of Γ0(N) in Γ0(1) and
FN as above.
Let w ∈ FN. We want to show that ℑ
σ −1
to Remark 1.1.1 it is enough to show that ℑ
,
I(w) Uww
√
3
≥ 2N
σ −1
j(ℓ) Uℓw
≥
, and according
√ 3
2N
for some ℓ.
Since w ∈ FN = ∪Vk(F1) there is an index k such that w ∈ Vk(F1) = Rk. Let
15

v ℓ(k) = Vk(∞) be the vertex at infinity of Rk, and p j(k) the cusp representative
equivalent to v ℓ(k), i.e. U ℓ(k)v ℓ(k) = p j(k).
Note that we can write σ −1
j = l js jω j where l j is a translation, ω j ∈ PSL(2,Z)
is such that ω jS jω −1
j = Sh j , (where S h z = z + h) and s j is a scaling with 1
h j ,
where the positive integer h j is called the width of the cusp p j (cf. [51, p.
36-37]). The width, h j is independent of the choice of cusp-representative in
the class of p j (cf. [63, p. 59]). Now ω j(k)U ℓ(k)Vk(F1) is simply a horizontal
translate of F1, so ℑ(ω j(k)U ℓ(k)w) ≥
√ 3
2
and thus ℑ(σ −1
j(k) U ℓ(k)w) ≥
[51, p. 36-37] it is clear that 1 ≤ h j ≤ N for all j, hence ℑ(σ −1
and accordingly Y0 ≥
√ 3
2N .
√
3
2h . From
j(k)
j(k) U √
3
ℓ(k)w) ≥ 2N ,
Remark 1.1.2. If P is a prime ≥ 3 we will see that for a natural choice (which √
3
we use in the numerical work) of fundamental domain we have Y0(P) = 2P .
Let T z = −1
z and Sz = z + 1. We know that the maps V1 = Id and Vk =
P+3
T S (k− 2 ) for k = 2,...P + 1 is a set of right coset representatives for Γ0(P)
in Γ0(1) (cf. [83, p. 135]). The corresponding fundamental domain is FP =
∪ P+1
k=1Vk(F1), it has one cusp of width 1 at p1 = ∞ and one cusp of width P at
p2 = 0. The cusp normalizing maps are σ1 = Id and σ2z = −1
1
Pz . Let e = 2 +i
√
3
2 .
√
3
Then we see that ℑ(VP+1(e)) = 2 1
√
3 P−1
2 + i 2 + 2 −2 √
3 4 = 2 P2 +3 ≤
√
3
2P (since
P ≥ 3), and clearly ℑ(σ −1
2 VP+1(e)) = ℑ √
e 1 1 3
P + 2 − 2P = 2P . Hence the height
√
3
of VP+1(e) is equal to 2P , which means that Y0
√
3
≤ 2P . The conclusion is that
for a prime level P ≥ 3 and for this particular choice of fundamental domain
we have Y0 =
√ 3
2P .
1.1.3 Introduction to Dirichlet Characters
First we fix m ∈ Z + . A Dirichlet character, χ mod m, is a group homomorphism
from (Z/mZ) ∗ to the unit circle S 1 , which is viewed as a function on Z
(with period m) by assigning χ(n) = 0 if (n,m) > 1. We then define the map
χ : SL(2,Z) → S 1 by
χ
a b
c d
= χ(d).
Note that χ = 0 on Γ0(m) since ad − bc = 1 and c ≡ 0 mod m imply that
χ(ad) = χ(a)χ(d) = 1. Observe also that χ is a group homomorphism from
Γ0(m) to S 1 .
If χ has a period less than m (for values of n restricted by (n,m) = 1) then
χ is said to be imprimitive, otherwise χ is said to be a primitive character.
16

(Note that the trivial character, χ(n) ≡ 1, is imprimitive for m ≥ 2). If q is the
smallest period of χ, then q is called the conductor of χ.
If χ has period m and conductor q there is a unique way to define a character
χ ′ mod m ′ , for all modulus m ′ which are multiples of q or multiples of m.
We say that the character χ is even if χ(−1) = χ(1) = 1, and odd if χ(−1) =
−χ(1) = −1.
Since we are actually concerned with PSL(2,Z) we need χ(A) = χ(−A) to
hold for any A ∈ SL(2,Z), hence we will only consider even characters.
It is known (see [27]) that primitive real characters exist only for moduli of
the following types:
m = N1,4N2, or 4N3,
where N1 ≡ 1 mod 4, N2 ≡ 2 mod 4 and N3 ≡ 3 mod 4 and N1,N2 and N3 are
square-free.
One should note that these are precisely the fundamental discriminants of
real quadratic fields, and it is also a fact that all real characters are given by
quadratic residue symbols (Kronecker’s extension of the Legendre symbol)
cf., e.g. [25, p. 37].
Since characters are multiplicative it is fairly easy to evaluate them at products
of integers, but when it comes to linear combinations of integers it is
harder. A useful trick for evaluating characters at certain linear combinations
is to factor the character instead of the integer. Here we will give the simplest
case, which will be of use for the proof of Proposition 1.2.6 in Subsection
1.2.8.
Fact 1.1.1. Suppose that χ is a real primitive character mod N and that m ∈
Z + is such that m|N and m, N
m = 1. Then it is known (cf. [69, (3.1.4)]) that
we can factor χ as χ = χmχ N , where χm and χ N are real primitive characters
m
m
mod m and N
m respectively.
Example 1.1.1. Let χ, N and m be as above, and let t = qA + N
q B for some
A,B ∈ Z. Then
χ(t) = χm(t)χ N (t) = χm mA +
m
N
m B
χ N mA +
m
N
m B
N
= χm
m B
χ N (mA).
m
Definition 1.1.5. A character χ defined on Γ0(N) is said to leave the cusp
p j open if it is trivial on the stabilizer subgroup Γp j , i.e. if χ(S j) = 1. If a
character leaves all cusps open we say that the character is regular for Γ0(N).
If a character does not leave the cusp open we say that the cusp is closed,
and it is known that there is no contribution to the continuous part of the spectrum
from a closed cusp (cf. [39, pp. 91-99]).
17

Observe that the cusp at ∞ is fixed by S : z ↦→ z + 1 so χ(S) = 1 for any
Dirichlet character. Since we know that S j = σ jSσ −1
j we will also have
χ(S j) = 1 whenever σ j is a (Γ0(N), χ)−normalizer as described in Definition
1.2.1 below. In particular, for square free N ≡ 1 mod 4 all Dirichlet characters
are regular (cf. [12]).
The basic assumption from now on (unless anything else is explicitly stated)
is that any character χ is a real and even Dirichlet character.
1.1.4 A Brief Introduction to Maass Waveforms
For a Fuchsian group Γ, a function f defined on the upper half-plane is called
Γ−automorphic if it is invariant or transforms in a chosen manner (i.e. with
respect to characters or general multipliers) under the action of Γ by the so-
called slash operator of weight k ∈ Z defined by
f |[k,A](z) =
cz + d
|cz + d|
−k
f (Az), A ∈ Γ.
Such functions can be viewed as “living” on the Riemann surface Γ\H (or
coverings of it in the case of characters). In this chapter we will only consider
certain non-holomorphic L 2 −functions, the so called Maass waveforms (to be
defined below) and only at weight k = 0 (note that f |[0,A](z) = f (Az)).
In the hyperbolic metric on H the Laplace-Beltrami operator, ∆, takes the
form
∆ = y 2
∂ 2 ∂ 2
+
∂x2 ∂y2
.
Definition 1.1.6. If Γ is a Fuchsian group, a Maass waveform for Γ is a function
f defined on H with the following properties
i) ∆ f + λ f = 0, λ ≥ 0;
ii) f (Az) = f (z), for all A ∈ Γ;
iii)
Γ\H | f |2 dµ < ∞.
We usually write the eigenvalue as λ = 1
4 + R2 . It is conjectured (Selberg)
that for congruence subgroups, if λ = 0, then we can take R ∈ [0,∞) (λ ≥
1
1
4 ); any eigenvalues λ ∈ (0, 4 ] are usually called exceptional. We denote the
space of Maass waveforms for Γ by M(Γ), and the space corresponding to a
particular eigenvalue by M(Γ,λ). If χ is a group homomorphism from Γ to
S1 (cf. Subsection 1.1.3) we can replace condition ii) by
ii’) f (Az) = χ(A) f (z), for all A ∈ Γ.
18

We then say that f is a Maass waveform for (Γ, χ), the space of which we
denote by M(Γ, χ) or M(Γ, χ,λ). It is also known that each space M(Γ, χ,λ)
is finite dimensional (see for example [39, p. 140, thm. 11.11]), that Maass
waveforms span the discrete part of the spectra of ∆, and that the continuous
part is spanned by the Eisenstein series (see for example [50, 97] or [39]).
It is an important fact that if Γ is a congruence subgroup then every Maass
waveform f in M(Γ, χ,λ) with λ > 0 is a cusp form, meaning that it tends
rapidly to 0 in each cusp. This fact follows from [39, pp. 327(note 15),
78(claim 9.6), 284(line 1), 328(line -2)] since the scattering matrix of Γ(N)
can be computed explicitly and shown not to have any poles for 1
2 < s < 1; cf.,
e.g., [75] or [48], and [69, p. 114].
We can equip the space M(Γ, χ,λ) with the Petersson inner-product,
< f ,g >= f ¯gdµ,
where the integration can be taken over any fundamental domain for Γ. With
this inner-product M(Γ, χ,λ) is now a finite dimensional Hilbert space.
If Γ is a congruence subgroup and χ is trivial on Γ(N), the eigenvalues 0 =
λ0 < λ1 ≤ λ2 ≤ ···, counted with multiplicity, form a discrete sequence and
a general Weyl’s law is known to hold. In particular, for Hecke congruence
subgroups and the trivial character, we have (see [86, thm. 2])
√
T
N Γ0(N)(T ) = µ(FN)
4π
as T → ∞,
Γ\H
2κ √ √ √
T − T ln T + A T + O
π
ln √ T
, (1.3)
where A is a certain constant depending on the level N and κ is the number of
cusps of Γ0(N).
However it is widely believed (by the Sarnak-Phillips philosophy, see for
example [80]) that for a “generic” non-cocompact but cofinite Fuchsian group
Γ, unless there are some arithmetic or geometric symmetries present, there
should only be at most a finite discrete spectra (generic here is in the sense that
in some appropriate deformation space the exceptional groups have measure
0).
The Maass waveforms are also more mysterious than the holomorphic automorphic
functions since there are only very few examples of explicit formulas
for constructing Maass waveforms, whereas there are numerous examples of
explicit formulas for holomorphic modular functions.
19

1.2 Some Structural Theory of M(Γ0(N), χ,λ)
For the rest of this section we put Γ = Γ0(N) for brevity and we also assume
that χ is a Dirichlet character. To facilitate the computation of Maass waveforms
we need as much information as possible about the various symmetries
that can be used. First we will see that in the case of a real character we can
assume that our functions are real-valued. Then we will consider the obvious
translational symmetry which makes the Fourier expansion possible, and
the reflectional symmetry which simplifies the said Fourier expansion. Then
we will describe the less obvious symmetries, the Hecke operators and the
involutions, which further refine the spectral eigenspaces.
1.2.1 The Conjugation Operator
Let f ∈ M(Γ, χ,λ) and consider the conjugation operator K, K f = f . It is
clear that K commutes with the Laplacian so K f is also an eigenfunction of ∆
with the same eigenvalue as f . We also see that for A ∈ Γ we get
(K f ) |A(z) = K f (Az) = f (Az)
= χ(A) f (z)
= χ(A)(K f )(z),
so K f is automorphic with respect to the conjugate of χ, i.e. K f ∈ M(Γ, χ,λ).
In particular, if χ is a real character, then ¯f = K f ∈ M(Γ, χ,λ). By considering
the functions 1
1
2 ( f + K f ) and 2i ( f − K f ) (which are real-valued) we get
the following proposition.
Proposition 1.2.1. If χ is a real-valued character then M(Γ, χ,λ) has a C−linear
basis consisting of real-valued functions.
1.2.2 The Fourier Series
Notation 1.2.1. In connection with Fourier series of functions in M(Γ, χ,λ)
we will use the following notation:
and
e(x) = e 2πix ,
κn(y) = κn(R,y) = √ yKiR(2π|n|y),
where λ = 1
4 + R2 . Observe that κ−n(y) = κn(y).
If f ∈ M(Γ, χ,λ), we define functions f j = f |σ j and we know that f j(z+1) =
f (σ jSz) = f (S jσ jz) = χ(S j) f j(z) = f j(z) if the character leaves the cusp p j
open.
20

Let λ > 0. By separating variables in the Laplace equation it follows that f j
has a Fourier expansion of the form ∑n=0 c j(n)κn(y)e(nx) (cf. [39], and recall
that f is necessarily a cusp form, cf. Section 1.1.4).
Proposition 1.2.2. If χ is regular for Γ then a function f ∈ M(Γ, χ,λ) admits
a Fourier expansion at each cusp of Γ, and these expansions are given by the
functions f j. Explicitly we have the following expansion at the cusp p j :
f j = f |σ j = ∑ c j(n)κn(y)e(nx), j = 1,...,κ.
n=0
1.2.3 Involutions and Normalizers
Definition 1.2.1. For Γ ⊆ PSL(2,R) we say that g ∈ PGL(2,R) is a normalizer
of Γ in PGL(2,R) if
gΓg −1 = Γ.
This means in particular that A ∈ Γ ⇒ ∃B ∈ Γ such that gA = Bg. But we
need a stronger definition if a character is present.
We say that g is a normalizer of (Γ, χ) if g is a normalizer of Γ and for all
A ∈ Γ:
χ(gAg −1 ) = χ(A).
The set of all normalizers of Γ in PGL(2,R) forms a group, the so-called
normalizer subgroup of Γ in PGL(2,R).
In the above definition we have PGL(2,R) = GL(2,R)/C, where C is the
center in GL(2,R) and consists of all diagonal matrices of the form
a 0
0 a with
a = 0.
We define an action of GL(2,R) on H as follows, for g =
a b
c d and z ∈ H:
⎧
az+b
⎪⎨ cz+d , if ad − bc > 0,
g(z) =
⎪⎩ az+b
cz+d , if ad − bc < 0.
Note that all of C acts trivially on H, and hence we also obtain a welldefined
action of PGL(2,R) on H. And if g is a normalizer of (Γ, χ) and
f ∈ M(Γ, χ,λ) then f |g ∈ M(Γ, χ,λ).
The usual definition of a (linear) involution of a vector space S is a linear
operator T : S → S such that T 2 = Id (or equivalently T −1 = T ).
Definition 1.2.2. A linear operator T : M(Γ, χ,λ) → M(Γ, χ,λ) will be called
a (Γ, χ)-involution if T 2 is the identity operator, that is
for any function f ∈ M(Γ, χ).
T 2 f = f
21

Example 1.2.1. There are three typical examples of (Γ, χ)−involutions that
we will use.
1. All W ∈ PSL(2,R) which are normalizers of (Γ, χ) and satisfy W 2 ∈ Γ and
χ(W 2 ) = 1 are (Γ, χ)-involutions when acting via f ↦→ f |W . In particular
we will use ωN : z ↦→ −1
Nz , which can be represented by
0 −1
√N
√ N 0
PSL(2,R).
2. If J =
1
0
0
−1
is a normalizer of (Γ, χ) then f ↦→ f |J is a (Γ, χ)involution.
Note that this is just a reflection in the imaginary axis, J(z) =
−¯z.
It is important not to confuse the cusp normalizing map defined in the beginning
with the normalizer of (Γ, χ) defined above. But some cusp normalizing
maps might be normalizers of (Γ, χ) and even involutions of M(Γ, χ). In general,
if it is possible, this is a very good choice of cusp normalizing map as we
will see.
1.2.4 The Reflection Operator
By looking at the fundamental domain of PSL(2,Z) (or the Ford fundamental
domain of any congruence subgroup) we see that there is an obvious symmetry;
reflection in the imaginary axis. This symmetry provides us with a
partitioning of the spectrum into even and odd functions and amounts actually
to looking separately at Dirichlet or Neumann boundary conditions when we
view the spectral problem on the fundamental domain itself (which has finite
boundary points) and not on the corresponding Riemann surface PSL(2,Z)\H
(with the only boundary point at the cusp).
As remarked in Example 1.2.1, the reflection operator is represented
by the
1 0
a b
matrix J =
, and for any matrix T = we define
0 −1
c d
T ∗ = JT J −1
1 0 1 0 a −b
=
T
=
. (1.4)
0 −1 0 −1 −c d
Then J is a (Γ, χ)−involution whenever T ∗ ∈ Γ and χ(T ∗ ) = χ(T ) for all T ∈
Γ. In particular this is clearly true for Γ0(N) when χ is a Dirichlet character
since then χ(T ∗ ) = χ(d) = χ(T ).
Suppose now that J is a (Γ, χ)−involution. We can diagonalize M(Γ, χ,λ)
with respect to J and the eigenvalues will be either 1 or −1. f ∈ M(Γ, χ,λ) is
said to be even or odd, respectively, if f |J = f or f |J = − f . Every even or odd
22
∈

function f in M(Γ, χ,λ) has a cosine resp. sine Fourier series. This can be
seen as follows. Let
f (z) = ∑ c(n)κn(y)e(nx)
n=0
with κn(y) as in Notation 1.2.1 on page 20 (remember that κ−n(y) = κn(y)).
Then
f |J(z) = ∑ c(n)κn(y)e(−nx)
n=0
= ∑ c(−n)κn(y)e(nx).
n=0
If f |J = f (−¯z) = ε f , then we get c(−n) = εc(n) so
f (z) = ∑ c(n)κn(y)e(nx)
n=0
=
=
=
∞
∑
n=1
κn(y)(c(n)e(nx) + c(−n)e(−nx))
∞
∑ c(n)κn(y)(e(nx) + εe(−nx))
n=1
∞
∑ c(n)κn(y)
n=1
2cos(2πnx), ε = 1,
2isin(2πnx), ε = −1.
Set a(n) = 2c(n) and b(n) = 2ic(n), then f will have a Fourier cosine or sine
series with coefficients a(n) or b(n) respectively, i.e.
f (z) =
f (z) =
∞
∑
n=1
a(n)κn(y)cos(2πnx), or
∞
∑ b(n)κn(y)sin(2πnx).
n=1
So we know that f = f0 can always be taken as an eigenfunction of J, but we
should also note that this need not be the case with the other Fourier series, f j.
In the next section we give a condition for the simultaneous diagonalization
of all f j with respect to J.
1.2.5 Complete Symmetrization
Since it is usually preferable to work with symmetrized Fourier series containing
only cosines or sines instead of the exponential functions we make a
comment on when this is possible.
23

Theorem 1.2. Let u
v be any cusp satisfying (u,v) = 1, v|N and v, N
v = 1 or
2. Then if we take p j = u
v as a cusp representative, there is a choice of cusp
normalizing map σ j such that if f ∈ M(Γ, χ,λ) is an eigenfunction of J with
eigenvalue ε then f j is also an eigenfunction of J with the eigenvalue χ(d)ε,
where d is unique modulo N determined by d ≡ 1 mod v and d ≡ −1 mod N
v .
Proof. (We refer to [51, p. 36] for the basic facts).
We know that every cusp for Γ has a representative of the form u
v where v|N
and (u,v) = 1, and u u′
v and v ′ are equivalent cusps if and only if v = v ′ and u ≡
u ′ mod v, N
u
u
v . Hence v is equivalent to − v if and only if 2u ≡ 0 mod v, N
v
and this is possible if and only if v, N
v = 1 or 2. Suppose now that this is
the case, and that we have p j = u
v for such choice of u, and that A =
a b
Nc d ∈
Γ0(N) is such that Ap j = −p j. One verifies by a straightforward computation
that up to a sign change (A ←→ −A), the most general form of A is given
by taking c as an arbitrary integer satisfying
N
v cu ≡ −2 mod v (thus c is
uniquely determined modulo v if v, N
v = 1, and uniquely determined modulo
v
2 if v, N
v = 2), and then letting
d = a = −1 − N
u(1 − d)
· cu, b = .
v v
In particular, note that d ≡ −1 mod
N and d ≡ 1 mod v.
v
Observe now that f |T |J = f |J|T ∗ (where T ∗ is as in (1.4)). This means in
particular that if f has J−eigenvalue ε, then f j|J = f |σ j|J = f |J|σ ∗ = ε f
j |σ ∗. We
j
can choose the cusp normalizing map of p j as
√m
u x
0
σ j =
v y 0 √ m −1
, (1.5)
for some integers x,y satisfying uy − vx = 1 and m = N
(N,v2 is the width of the
)
cusp p j (note that m = N N
v or 2v if v, N
v = 1 or 2). This choice of σ j is unique
up to right multiplication with Sr for some r ∈ R. Then we will have
σ ∗ j = Jσ jJ −1 √m
u −x
0
=
−v y 0 √ m −1
,
which is clearly a cusp normalizing map for −p j. Since Ap j = −p j, then
another cusp normalizing map of −p j is the map Aσ j, and hence we have
σ ∗ j = Aσ jSt for some t ∈ R.
It follows that f j|J = ε f |σ ∗ = ε f
j |Aσ jSt = χ(d)ε f j|St , and hence if t ∈ Z we
have f j|J = χ(d)ε f j, as desired.
Hence it only remains to make sure that we can obtain t ∈ Z above. In general
this can always be achieved by replacing σ j by ˜σ j where σ˜ j = σ jS t 2 , since
24

then ˜σ ∗ j = σ ∗ j S− t 2 = Aσ jSt S− t 2 = Aσ jS t 2 = A ˜σ j. However, if either v, N
v = 1
or m is odd, then we can take x and y to be integers in (1.5). To see this, note
that by writing out the upper right element in the matrix relation σ ∗ j = Aσ jSt (and using au + bv = u) one obtains −x = umt + ax + by. Using the formulas
for a and b this implies that mt = N d−1
v cx+ v y. Hence t ∈ Z holds if and only if
N d − 1
cx + y ≡ 0 mod m. (1.6)
v v
Recall that m = N N
1−d
v or 2v , so that (1.6) is equivalent to v y ≡ 0 mod m. Hence
we achieve our goal if we let x,y be any integers satisfying uy − vx = 1 and
y ≡ 0 (mod m). This is clearly possible if either v, N
v = 1 or m is odd, since
then (v,m) = 1. (In the remaining case, i.e. v, N
N
v = 2 and m = 2v is even, one
can check that one must allow x or y to be non-integral to obtain t ∈ Z.)
Corollary 1. Suppose that N can be written as N = 2r p1 ··· pm, with 0 ≤ r ≤ 3,
and if m > 0 then p1,..., pm are distinct odd primes, and suppose also that
in every character decomposition χ = χvχ N , v|N, the character χ N is even.
v
v
Then we can take all f j as simultaneous eigenfunctions of J with the same
eigenvalue.
Note that if we do not have simultaneous eigenfunctions of J we have to
work (numerically) with exponentials instead of sines and cosines, and complex
instead of real Fourier coefficients.
1.2.6 Hecke Operators
It is well-known that the classical theory of Hecke operators (cf., e.g. [5], [36]
etc.) can be carried over from the case of holomorphic modular forms to the
case of Maass waveforms. This section provides an outline of the theory for
those not familiar with Hecke theory, and it will also serve as a recollection of
fundamental facts and a common ground of notation for the more experienced.
The Hecke operators considered here are operators acting on spaces of modular
functions (i.e. functions automorphic with respect to congruence subgroups).
We will define the Hecke operators in a similar way as in Atkin-Lehner [5].
Let N ∈ Z + be given. For a prime p there is a subgroup of finite index in
Γ0(N),
a b
Γ0(N, p) =
∈ Γ0(N)
c d
b ≡ 0 mod p .
For any d ∈ Z + , we define the map Ad : z ↦→ dz in PSL(2,R). For d = p
we then have A −1
p Γ0(N, p)Ap = Γ0(N p) ⊆ Γ0(N) and it is easy to see that
25

χ(A −1
p BAp) = χ(B) so that Ap gives a map between the spaces of Maass waveforms
A −1
p : M(Γ0(N), χ) → M(Γ0(N, p), χ), (1.7)
f ↦→ fp = f |A −1
p .
We then want to map fp to a function once again in M(Γ0(N), χ) but in a
non-trivial way (i.e. not via Ap). Let µ
R j be the set of right coset repre-
j=1
sentatives of Γ0(N, p) in Γ0(N) used in [5, lemma 5]. R j will then have the
lower right entry d ≡ 1 mod N. If V is any element in Γ0(N), then for each
j there exists a unique i such that R jV = gRi for some g ∈ Γ0(N, p), and different
j’ s give different i’s. Using our special choice of R j one also checks
that χ(V ) = χ(g) in this relation. Thus for any function h ∈ M(Γ0(N, p), χ)
it is clear that the function ∑ µ
j=1 h |R j belongs to M(Γ0(N), χ). Hence we may
define the operator Tp : M(Γ0(N), χ) → M(Γ0(N), χ) as follows:
Tp f = 1
√ p
µ
∑ fp|R j
j=1
.
The factor of 1
√p is a convenient normalization. Working out the coset representatives
explicitly (see [5, lemma 5]) gives the following formula for a
prime number p with (N, p) = 1
Tp f (z) = 1
p−1
√
p ∑ f
j=0
z + j
p
Tq f (z) = Uq f (z) = 1 √ q
+ 1
√ p χ(p) f (pz),
and for prime q with q|N (compare, e.g., [69, p. 142 (4.5.26)]):
z + j
.
q−1
∑ f
j=0
We will follow the convention to call Tq with q|N an exceptional Hecke operator
and denote it by Uq instead of Tq. In principle, due to the multiplicative
nature of the Hecke operators (cf. Theorem 1.3 below) it is sufficient to define
Hecke operators for primes, and then use the multiplicativity (cf. 1.9 below
and [69, lemma 4.5.12]) to recursively define Tn for any positive integer n
with (n,N) = 1. To use the above construction for a composite number n is
quite elaborate and it is not done this way in the literature. Hecke (cf. [36, nos.
32, 35, 36, 37 and 41, p. 859]) does not explain how he arrives at the exact
definition of the operators Tn. To get a motivation for the formula (1.8) below
(including the character) it is easiest to use the approach of double cosets as
in Shimura ([89]) or Miyake ([69]). Using either of these constructions yields
the following definition.
26
q

Definition 1.2.3. For any n ∈ Z + and any f ∈ M(Γ0(N), χ,λ) we define the
Hecke operator Tn by the formula
Tn f (z) = 1
az + j
√
n ∑
. (1.8)
d
ad=n
d>0
d−1
χ(a) ∑ f
j=0
Theorem 1.3. The Hecke operators Tn with (n,N) = 1 are endomorphisms of
the space M(Γ0(N), χ,λ) which have the following properties:
TnTm = ∑
d|(m,n)
d>0
χ(d)Tmn , (1.9)
d2 for any integers n and m with (n,N) = (m,N) = 1, and (for the adjoint operator)
T ∗
n = χ(n)Tn. (1.10)
In particular
Tn
(n,N) = 1 forms a commutative family of normal operators
which also commutes with the Laplacian and the reflection J.
Proof. To prove the multiplicative property cf. [89, eq. (3.5.10)] and look at
the action of Tn on the Fourier coefficients of a Maass waveform f . Suppose
that c(k) and b(k) are the respective Fourier coefficients of f and Tn f , then the
following relation holds:
b(k) = ∑
ad=n,a,d>0
χ(a)c
kd
, (1.11)
a
with the usual convention that c(r) = 0 if r /∈ Z. We prove the equality (1.9)
by comparing the action on the Fourier coefficients of the left hand side and
the right hand side.
Next, to prove (1.10), a quick computation using (1.9) shows that it suffices
to treat the case n = p a prime. To prove that T ∗ p = χ(p)Tp one uses the
following relation between the Peterson inner products on the two groups:
< Tp f ,g > Γ0(N)= 1
√ p < fp,g > Γ0(N,p) .
This relation is an easy extension of [5, lemma 12] to the case of non trivial
character using the fact that χ(R j) = 1 for all right coset representatives R j of
Γ0(N, p) in Γ0(N) as given in [5, lemma 5]. Alternatively: see [51, thm. 6.20]
and [69, thm. 4.5.4(1)]. It is easy to show that Tn commutes with ∆ and J.
Remark 1.2.1. Actually ([69, thm. 4.5.13]) we also have
UqTn = TnUq,
27

for q|N and (n,N) = 1 but in general the operators Uq are not normal so they
are put aside for a while, until the next section, where we will introduce the
space of newforms.
Theorem 1.4. There exists an orthogonal basis {φ j} in M(Γ0(N), χ,λ) consisting
of eigenfunctions to all Hecke operators Tn with (n,N) = 1.
Proof. Observe that M(Γ0(N), χ,λ) is a finite dimensional vector space (see
[39, pp. 140, 298]) so we can choose an ON basis { f j} m j=1 and represent the
operators Tn by matrices Λ(n) in this basis. From Theorem 1.3 above it is
clear that Λ(n)
(n,N) = 1 is a family of commuting normal m × m matrices.
From elementary linear algebra arguments (the spectral theorem or [84,
thm. 9.4.1]) we know that there exists a basis {φ j} m j=1 where each φ j is an
eigenfunction of all Λ(n), and this is exactly what we want. Compare: [51,
thm. 6.21] and [69, thm. 4.5.4(3)].
1.2.7 Oldforms and Newforms
From (1.11) it is clear that if f has Fourier coefficients c(n) and Tn f = λ(n) f ,
then
c(n) = λ(n)c(1).
We would like to make sure that c(1) = 1 here, but unfortunately as we will
see below, it may happen that c(1) = 0.
Suppose that χ is an imprimitive character mod N of conductor q. If q|M
and Md|N for some positive integers M,d, let χM be the character induced
from χ on Γ0(M). Then
f (z) ∈ M(Γ0(M), χM,λ) ⇒ f (z), f |Ad (z) ∈ M(Γ0(N), χ,λ)
(with Ad as in section 1.2.6, cf. (1.7)), which is clear from the fact that
AdΓ0(N)A −1
N
d = Γ0 d ,d
N ⊆ Γ0 d ⊆ Γ0(M), and χ(AdBA −1
d ) = χ(B).
Definition 1.2.4. Let f ∈ M(Γ0(N), χ,λ). We say that f is an oldform if there
exist positive integers M and d with Md|N, a character χM mod M and a
function f1 ∈ M(Γ0(M), χM,λ) such that we can write f = f 1|A d . The smallest
such integer M is called the conductor (or level) of f . Compare: [51, p. 107
(bottom)] and [69, p. 162 (bottom)].
28
Suppose now that f is an oldform with Fourier series given by
f (z) = ∑ c(m)
m=0
√ yKiR(2π|m|y)e(mx).

Then
f |Ad (z) = f (dz) = ∑ c(m)
m=0
dyKiR(2π|m|dy)e(mdx)
= √ d ∑
n≡0 mod d
c
n
√yKiR(2π|n|y)e(nx).
d
Hence if d > 1 then the first Fourier coefficient of f (dz) is zero. This is an
example of one of the inconveniences of oldforms.
Let M old (Γ0(N), χ,λ) be the linear space spanned by the oldforms and then
define M new (Γ0(N), χ,λ) as the orthogonal complement with respect to the
Petersson inner product. We then have
M(Γ0(N), χ,λ) = M old (Γ0(N), χ,λ) ⊕ M new (Γ0(N), χ,λ).
It is easy to see that both subspaces are stable under the Hecke operators Tn
with (n,N) = 1, since T ∗
n = χ(n)Tn.
Definition 1.2.5. A function f ∈ M new (Γ0(N), χ,λ) is called a normalized
newform of level N if f is an eigenfunction of J and all Hecke-operators Tn,
(n,N) = 1, and the first Fourier coefficient of f is 1, i.e. c(1) = 1.
Proposition 1.2.3. If χ is a primitive character mod N and q|N, then the operator
Uq is unitary on M(Γ0(N), χ,λ); i.e.
U ∗ q = U −1
q .
Proof. The proof is elementary (at least for prime indices N) and goes by first
proving that U ∗ q = ωNUqω −1
N and then using properties of the Petersson innerproduct.
Here ωN is defined as in Example 1.2.1. (For a sketch of the proof
see [12]).
The following theorem is crucial to the theory of oldforms/newforms. See
Atkin-Lehner [5] and Miyake [69] for similar results in the classical setting.
For a more general adelic version, see Miyake [68]. The following version for
Maass waveforms is in principle a direct adaptation of [94, thm. 4.6] to the
case of non-trivial character.
Theorem 1.5. Let χ be a Dirichlet character with conductor q. The spaces of
newforms have the following properties.
(a) For each multiple N of q, the normalized newforms of level N and
character χ, mN Fj , form an orthogonal basis in the newspace
j=1
Mnew (Γ0(N), χ,λ). If we want to stress the level and/or character
we write Fj = F (N,χ)
j . This is called the Hecke-(eigen)basis of
Mnew (Γ0(N), χ,λ).
29

(b) Each Fj is an eigenfunction of all Tn with n ∈ Z + .
(c) A basis in the space M(Γ0(N), χ,λ) is given by
F (M,χ)
i (dz)
d,M ∈ Z+
,q|M ,dM|N, 1 ≤ i ≤ mM .
(d) Let χ ′ be a Dirichlet character mod M. If f ∈ M new (Γ0(N), χ,λ)
and g ∈ M new (Γ0(M), χ ′ , µ) with λ,µ > 0, then either there exists
an infinite number of primes p for which the Hecke eigenvalues of
Tp are different for f and g, or else N = M, λ = µ, χ ′ = χ and
f ≡ g.
Proof. Observe that the only thing that differs from [94, thm. 4.6] is that we
have a non-trivial character.
The proof of (a)-(c) can be done exactly as in the proof of [94, thm. 4.6]. The
proof of “multiplicity one”, (d) is a bit more complicated to extend. The main
difference from the case of trivial character is the fact that ωN (cf. Example
1.2.1) and Tn no longer commute and we need the Euler products of both f
and ωN f for the proof of Lemma 4.12 in [94], however it is a simple matter
to do the same thing as in the proof of [69, thm. 4.6.19], noting that on the
newspace both Tp and Uq are normal so any eigenform of all Hecke operators
Tn is also an eigenform of all T ∗
n and U ∗ q (i.e T ∗
n = ω −1
N TnωN and U ∗ q = U −1
q on
the newspace).
Remark 1.2.2. Note that Uq is not normal on the full space M(Γ0(N), χ,λ)
(unless, of course, if χ is primitive, in which case there are only newforms) so
therefore we would have no multiplicity one theorem if we tried to diagonalize
the full space with respect to all Hecke operators.
Theorem 1.6. Let f be a normalized newform with Tn f = λ(n) f and f |J = ε f .
If the Fourier expansion of f is
(where c(1) = 1) then
f (z) = ∑
|m|≥1
c(m) √ yKiR(2πmy)e(mx)
c(m) = λ(m), and
c(−m) = ελ(m),
for all m ∈ Z + . We also have the following multiplicativity relations
30
c(m)c(n) = ∑
d|(m,n),d>0
χ(d)c
mn
d 2
c(m)c(p) = c(mp), p|N, m ∈ Z.
, (n,N) = 1, m ∈ Z, (1.12)

Proof. The first part follows by adding f and f |J and reordering the sum. The
second part follows at once from the multiplicative relation in Theorem 1.3
which then apply to the Hecke eigenvalues λ(n), and when we set c(1) = 1 in
the relation c(n) = λ(n)c(1) we are done.
Proposition 1.2.4. Suppose that f ∈ M new (Γ0(N), χ,λ) is a normalized newform,
and that q is a prime such that q|N, then the following holds.
(a) If χ is primitive then |c(q)| = 1.
(b) If χ is trivial and q 2 ∤ N then c(q) = λq
√q , where λq = ±1. In fact
λq is the eigenvalue of −Wq, cf. Subsection 1.2.8 below.
(c) If χ is trivial and q 2 |N, then c(q) = 0.
Proof. For (a), use Prop. 1.2.3. The assertions (b) and (c) are the analogs of
[5, thm. 3]. Compare: [69, thm. 4.6.17 and and cor. 4.6.18(2)], and [84, thm.
9.4.8].
Recall that by Theorem 1.5 we can always choose a Hecke eigenbasis in the
newspace M new (Γ0(N), χ,λ) where the functions are normalized by c(1) = 1.
We stress that if the character χ is non-trivial, then we can not in general
assume that the Hecke basis is real valued. For by Theorem 1.3 we have
T ∗
n = χ(n)Tn whenever (n,N) = 1, and thus
λ(n) = χ(n)λ(n) (1.13)
and c(n) = χ(n)c(n). Hence, in particular, if χ(n) = −1 then c(n) is purely
imaginary, whereas if χ(n) = 1 then c(n) is real.
Remark 1.2.3. Suppose that χ is real and that f is a normalized newform in
the space M new (Γ0(N), χ,λ) with f |J = ε f and Hecke eigenvalues λ(n). Then
ε ¯f is also a normalized newform in M new (Γ0(N), χ,λ), but with Hecke eigenvalues
λ(n). Now f and ε ¯f are in general linearly independent since λ(n) =
λ(n) whenever χ(n) = 1 and λ(n) = 0. Thus the newspace M new (Γ0(N), χ,λ)
is in general multidimensional.
An exception to the above fact are the CM-type forms à la [46] (originally
constructed by Maass in [62]). These forms are real and (hence) they have
c(n) = 0 for all n ∈ Z + with (n,N) = 1 and χ(n) = 1.
31

1.2.8 The Cusp Normalizing Maps as Normalizers of (Γ0(N), χ)
Suppose that Q|N and = 1. Then following [5] and [6] we can attach
Q, N
Q
a normalizer of Γ0(N) to Q, call it WQ . This can be taken of the form
WQ =
Qx
Nz
y
Qw
,
where x,y,z,w ∈ Z, x ≡ 1 mod N
Q , y ≡ 1 mod Q and Q2xw − Nzy = Q. One
can also verify that WQ is a normalizer of (Γ0(N), χ) for certain χ (for examples
see below). A remarkable fact about WQ is that as an operator on
M(Γ0(N), χ,λ) it is independent of the choices of x,y,z and w. In fact, if
W ′ Q =
Qx ′ y ′
for any x ′ ,y ′ ,z ′ ,w ′ ∈ Z and detW ′ Q = Q, then (as operators)
Nz ′ Qw ′
W ′ Q = χQ(y ′ )χ N Q
any f ∈ M(Γ0(N), χ,λ)
(x ′ )WQ (see [6, prop. 1.1]). In particular, this implies that for
f |J|WQ = χQ(−1) f |WQ|J.
For us the normalizer WQ will be most useful for real characters because of
the following proposition (cf. [6, prop. 1.1]).
Proposition 1.2.5. Suppose that Q|N and Q, N
Q = 1, and let χ = χQχ N .
Q
Then for any f ∈ M(Γ0(N), χ,λ) we have WQ f = f |WQ ∈ M(Γ0(N), χQχ N ,λ),
Q
and W 2 Q f = χQ(−1)χ N (Q) f . In particular, if χQ is real and χQ(−1)χ N (Q) = 1
Q
Q
then WQ maps M(Γ0(N), χ,λ) into itself and WQ is a (Γ, χ)−involution.
The normalizer WQ can in many cases be taken to be a cusp normalizing
map, as explained in the following proposition.
Proposition 1.2.6. Suppose that Q|N and = 1. Then we can choose
WQ as a cusp normalizing map of the cusp Q
N .
Q, N
Q
In reference to Definition 1.1.4, note that WQ is equal to Q − 1 2WQ ∈ PSL(2,R).
Proof. We may choose x = z = 1 in the formula for WQ, that is
WQ =
Q
N
y
wQ
,
where y,w ∈ Z, and Qw − N
Q
Qy = 1. Note that the width of the cusp N is
32
N,
N
N
Q
2 =
N
Q
N
Q, N
Q
= Q

can be written
uniquely up to a translation as σ = Aρ, where A ∈ SL(2,Z) maps ∞ to Q
N , and
ρ is a scaling by the width of the cusp. Hence, writing WQ as
1 y Q 0
WQ =
,
0 1
(cf. [51, p. 37]). We know that the cusp normalizing map of Q
N
N
Q wQ
we see that WQ is a cusp normalizing map for Q
N .
Another interesting property of WQ is the following (cf. [6, prop. 1.2, 1.3]).
Proposition 1.2.7. Let Q ∈ Z +
be such that Q|N and
and p be primes with (p,N) = 1, q|N and (q,Q) = 1. Then
TpWQ f = χQ(p)WQTp f , and
UqWQ f = χQ(q)WQUq f ,
Q, N
Q
= 1, and let q
for f ∈ M(Γ0(N), χ,λ), where χQ is the character mod Q induced by χ (i.e. χ =
χQχ N ).
Q
Observe that for a Hecke eigenfunction f we get Tp(WQ f ) = χQ(p)WQTp f =
χQ(p)λpWQ f where λp is the Hecke eigenvalue of f for Tp. In other words
WQ f is also an eigenfunction of Tp with eigenvalue λpχQ(p). In view of “mul-
tiplicity one” (cf. Thm. 1.5 (d)), if χQ ≡ 1 and f is a newform, then WQ f must
be a multiple of f , i.e. WQ f = µQ f , where µ 2 Q = χ N (Q) (cf. Prop. 1.2.5).
Q
Now if we choose the cusp normalizer σ j of Q
N as σ j = WQ (cf. Prop. 1.2.6),
then we have f j = µ j f1 , and in terms of the different Fourier expansions, this
means that (for all k)
c j(k) = µ jc1(k), (1.14)
where µ 2 j = χ N (Q). This relation can either be used as a means to test the
Q
accuracy of our program or to reduce the size of the linear system in Section
1.3.2 (cf. the normalization (B) on p. 38). However, it is important to observe
that (1.14) does not in general allow us to reduce the Fourier coefficients at all
cusps to the ones at i∞. Cf. Prop. 1.2.6.
Our comment in Prop. 1.2.4(b) about −Wq is justified by combining the
foregoing remarks with the three references cited in the proof of Prop. 1.2.4.
Cf. [6, §1,2], [51, pp. 113–118] and [69, cor. 4.6.18] for some related ideas.
33

1.3 Computational Aspects
1.3.1 Introduction
We know that a Maass waveform f ∈ M(Γ0(N), χ,λ), with λ > 0 is completely
described by its Fourier series at ∞, but unfortunately to assure stability
of the numerical method we need knowledge of the Fourier series at all
cusps of F, i.e. for 1 ≤ j ≤ κ,
f j(z) = ∑ c j(n)κn(y)e(nx).
|n|≥1
The aim here is to compute such functions f , that is, we wish to find an eigenvalue
λ = 1
4 +R2 (for convenience we will usually refer to R as the eigenvalue)
and a set of Fourier coefficients {c j(n)|1 ≤ j ≤ κ, |n| ≥ 1}. In general (the
exceptions are the CM-type forms, cf., e.g., [46]) there are no known formulas
for neither the eigenvalues nor the coefficients and we have to be content with
numerical approximations.
There are two steps (Phase 1 and Phase 2) in the algorithm. First we locate R
(up to the desired accuracy) and an initial set of Fourier coefficients, and then,
if we want to, we can use this initial set to generate a larger set of coefficients.
1.3.2 Phase 1
The method here is a generalization of Hejhal’s algorithm in [43] to groups
with several cusps and non-trivial characters, and some of the ideas are also
inspired by the algorithm by Selander and Strömbergsson in [95]. The idea is
that given a real number R we use linear algebra to compute a set of numbers
that are likely to be close to “true” Fourier coefficients if R is close to a “true”
eigenvalue of a Maass waveform.
Preliminary Numerical Remarks
First we introduce an “effective zero”. In standard double precision arithmetic
we know that x + 10 −16 x ≈ x, so if we fix ε < 10 −16 then a truncation error of
ε can in principle be neglected (computationally). We will use [[ε]] to denote
a quantity with absolute value less than ε.
Suppose that f ∈ M(Γ0(N), χ,λ) with Fourier expansions at the cusps
f j(z) = ∑ c j(n)κn(y)e(nx), 1 ≤ j ≤ κ. (1.15)
|n|≥1
There is a trivial (a priori) bound on the coefficients: c j(n) = O( √ n) (cf. [39,
p. 585] and [50, thm. 3.2]). Combining this with κn(y) = √ yKiR (2π|n|y) and
the fact that KiR(u) ∼ π
2ue−u , as u → ∞ (for fixed R), we see that the tail
of the sum (1.15) satisfies ∑|n|≥M = O e−2πyM as M → ∞, with the implied
34

constant depending on R and y. Hence for any fixed y (and R) we can take
M = M(y) such that
f j(z) = ∑
1≤|n|≤M(y)
c j(n)κn(y)e(nx) + [[ε]], (1.16)
for 1 ≤ j ≤ κ. Using similar observations as in [43, p. 311] it is clear that, for
R large, we can take M(y) as
R + AR
M(y) =
1
3
, (1.17)
2πy
for some constant A. In practice it turns out that A = 12 or 15 is good enough.
We will now see how to use the automorphy condition ((1.19) below) to get a
good linear system that can be solved for the c j(n).
The Linear System
In order to make the algorithm stable we need to use the Fourier coefficients
at all cusps, which means that we have to use different expansions at different
regions in a clever way as follows (see also [95]). Recall the notation from
Subsection 1.1.2.
We will view f ∈ M(Γ0(N), χ,λ) as pieced together of the Fourier series
at all cusps, meaning that we use the f j that is most convenient at each point.
Hence, given w ∈ FN we will use the identity
f (w) = χ(U −1
w ) f I(w)(σ −1
I(w) Uww). (1.18)
Cf. Remark 1.1.1 on p. 15.
Given any point z ∈ H and any j ∈ {1,...κ}, we let z j = σ jz (which may
or may not be inside FN), and let w j be the pullback of z j into FN, i.e. w j =
Tj(z j) ∈ FN with Tj ∈ Γ0(N), and let z∗ −1
j = σI( j) Uw jw j (here I( j) = I(w j)), then
z∗ −1
j = σI( j) Uw jTjσ jz, so
f j(z) = f (σ jz) = f (T −1 −1
j Uw j σI( j)z ∗ j)
= χ(T −1 −1
j Uw j ) fI( j)(z ∗ j). (1.19)
This relation (implicit automorphy, cf. [43, p. 298 (15)]) between f j(z) and
fI( j)(z∗ j ) is what enables us to get hold of all Fourier series. Consider the
truncated Fourier series
ˆf j(z) = ∑
1≤|n|≤M(y)
c j(n)κn(y)e(nx).
One way to view this series is as a Discrete Fourier Transform, and we can
perform an inverse transform over the following set of sampling points along
35

a horocycle:
zm = xm + iY
xm = 1
1
(m − ), 1 − Q ≤ m ≤ Q ,
2Q 2
for some Y < Y0 (recall (1.2) on page 15), and Q > M(Y ). The inverse transform
gives us that for 1 ≤ |n| ≤ M(Y ) < Q we have
c j(n)κn(Y ) = 1
2Q
= 1
2Q
Q
∑
m=1−Q
Q
∑
m=1−Q
ˆf j(zm)e(−nxm)
f j(zm)e(−nxm) + [[ε]],
where in the last step we used (1.16). This system is now almost a tautology,
but we can use the implicit automorphy (1.19) to get a good “mix” of the
Fourier coefficients and a far from trivial system.
Mimicking the the discussion leading to (1.18) we let Tm j ∈ Γ0(N) be the
pullback map of σ jzm into FN, and let wm j = Tm j(σ jzm) ∈ FN, I(m, j) =
−1
= σI(m, j) Um jwm j. Furthermore we let χm j =
). Using (the analog of) (1.19) we then obtain:
I(wm j), Um j = Uwm j ∈ Γ0(N), z∗ m j
χ(T −1 −1
m j Um j
c j(n)κn(Y ) = 1
2Q
Q
∑ χm j fI( j,m)(z
m=1−Q
∗ m j)e(−nxm) + [[ε]],
and we now want to substitute the truncated Fourier expansion for fI( j,m).
Since we know that ℑ(z∗ −1
m j ) = ℑ(σI( j,m) Um jTm jσ j(zm j)) ≥ Y0 we can actually
use the same truncation point, M0 = M(Y0) for all series, and in interchanging
the order of summation we get the following expression, valid for 1 ≤ |n| ≤
M(Y ) < Q and 1 ≤ j ≤ κ :
where
c j(n)κn(Y ) =
V ji 1
nk =
2Q
Q
∑
m=1−Q
I( j,m)=i
κ
∑
∑
i=1 1≤|k|≤M0
ci(k)V ji
nk + 2[[ε]], (1.20)
χm jκk(y ∗ m j)e(kx ∗ m j)e(−nxm). (1.21)
Remark 1.3.1. It seems clear that in order for the system (1.20) to be useful,
we need to use the automorphy relation non-trivially sufficiently often. In
practice we found that a necessary (and sufficient!) condition for the numerics
to behave well is that z ∗ m j = zm for all j,m. This condition is ensured if we
36

keep ℑ(z∗ m j ) > Y for all j,m. This is of course automatically fulfilled if we
choose Y < Y0 as above, but in many cases it is possible to verify numerically
that the same inequality also holds for certain choices of Y (slightly) larger
than Y0. This means that we may be able to use a Y > Y0 and a corresponding
M(Y ) < M0. The drawback is that such a Y can not be used for Phase 2 later
since there the pullbacks z∗ m j of the horocyclic points zm will eventually fill out
the whole fundamental domain (as Q → ∞ and Y → 0 simultaneously).
We now have a linear system that can be used to obtain the coefficients.
Note that the V ji
nk can be small due to either the K-Bessel decay or “bad mixing”
in the sense that the numbers of m such that I( j,m) = i might differ
much (meaning that the pullback of the horocycle does not encircle each cusp
equally much). Bad mixing is avoided basically through increasing the length
of the horocycle by means of decreasing Y . The system (1.20) can be expressed
as
0 =
κ
∑
∑
i=1 |k|≤MY
ci(k) ˜V ji
nk + 2[[ε]], (1.22)
where ˜V ji ji
nk = Vnk − δnkδ jic jκn(Y ). The term −κn(Y ) occurring in all diagonal
entries gives us good reason to hope that this system should turn out to be
well-conditioned, just as in [43, p. 298 (19) and second paragraph of p. 299]).
We now have a linear system which is satisfied by (linear combinations of)
Fourier coefficients of Maass waveforms f ∈ M(Γ0(N), χ,λ) (up to the error
2[[ε]]). If we let V denote the (κM0 × κM0)−matrix ˜V ji
nk and C denote the
κM0−vector of Fourier coefficients c j(k), we can write the system as
VC = 0. (1.23)
Observe that the solution space of this system for a true eigenvalue R is at least
a 1-dimensional linear space, so in order to get a unique solution we need to
use some sort of normalization.
Of course, if f and all the various f j are all Fourier cosine series, there is
an immediate “cosine” counterpart of (1.23). Similarly for sine series. This
remark is important in all cases where Cor. 1 applies. The matrix coefficients
in these analogs of (1.23) are all real!
Normalization
Depending on the dimension of the particular space M(Γ0(N), χ,λ) we are
investigating, we have to employ different types of normalizations algorithmically.
In order to test the accuracy of our coefficients (cf. p. 41 below) we
will always try to get eigenfunctions of Hecke operators and/or involutions
whenever possible. We will now discuss two examples (A) and (B) of possible
normalizations.
37

(A) Newforms, trivial character. According to a standard conjecture (and
experimental findings) the space of newforms with trivial character should
have dimension 1. So, by Theorem 1.5 we can search for a normalized newform
by setting c1(1) = 1 and dropping the first equation from V . If we find a
true R-value under this assumption, the coefficients will automatically satisfy
all multiplicative relations (cf. Theorem 1.6).
(B) N and real primitive χ satisfying the hypotheses of Cor. 1. To discuss
this case some preliminaries are necessary. We present them in outline form.
1) Let MR(Γ0(N), χ,λ) ⊆ M(Γ0(N), χ,λ) be the space of all real-valued
Maass waveforms. By simple use of the operator K, one sees that
M(Γ0(N), χ,λ) = MR(Γ0(N), χ,λ) + iMR(Γ0(N), χ,λ)
as vector spaces. The operators WQ and Tn are all endomorphisms on the
spaces M(Γ0(N), χ,λ) and MR(Γ0(N), χ,λ) (see Prop. 1.2.5 and Thm. 1.3).
One knows that W 2 Q = ±Id, and that TnWQ = χQ(n)WQTn for (n,N) = 1 (can
be proved by induction on Prop. 1.2.7.)
3) When χ is primitive, there are no oldforms. Cf. Thm. 1.5. Thm. 1.5(d)
assures us of multiplicity one for any newforms.
4) Apply Thm. 1.4 to M(Γ0(N), χ,λ) and get an orthogonal basis F1,...,Fm
consisting of normalized newforms. (Do not confuse things here with (1.15).)
By Thm. 1.6 and Cor. 1 each Fj is either a cosine or sine series at infinity and
remains of the same type at the other cusps.
5) It is now natural to apply Remark 1.2.3 to the basis functions Fj. If we
form the the normalized newform εFj there are two possibilities.
a) We arrive back at Fj. This is equivalent to saying λ(n) = λ(n), for all n.
Since λ(n) = χ(n)λ(n), for (n,N) = 1, by (1.13), this would imply λ(n) = 0
whenever χ(n) = −1.
b) We arrive at a form independent of Fj. By mult. 1, we thus have that
the sequences λ(n) and λ(n) are essentially different as n → ∞. Accordingly,
here, via (1.13), we have λ(n) =nonzero pure imaginary for infinitely many
primes with χ(n) = −1. While, λ(1) = 1, of course.
6) We apply 5) to each Fj, and thus deduce a natural splitting of the basis
{F1,...,Fm} into singlets and pairs.
7) In pursuing the numerics on the machine, we hypothesize (at least initially)
that either m = 1 (singlet) or m = 2 (one pair) for our given space
M(Γ0(N), χ,λ). In the following comments, we assume ε = 1. The treatment
of ε = −1 is similar.
8) Take m = 1 first. Look at 1
2F1. This is a unit normalized real cosine
series. Its expansions at the other cusps are again real cosine series. Take the
cosine analog of (1.23) and declare λ(1) = 1. Solve the system in the usual
way with two parallel Y. This case should have a unique solution.
38

Cases where χ(n) = −1 should yield coefficients that are exactly 0. This
will be very compelling to see! (Cf. Table 1.5, right column.)
9) Now assume m = 2. The (complex) basis is {F1,F2}, where F2 = εF1 =
F1 is the “Remark 1.2.3 flip” of F1. The idea here is to construct a real basis
from F1 and F2. To this end we construct F+ = 1
4 (F1 + F2) and F− =
1
4i (F1 − F2). These functions are now real cosine series at all cusps. F+ is
unit normalized, while the n = 1 term for F− is zero. Observe that the Fourier
coefficients of F+ and F− are ℜ(λ(n)) and ℑ(λ(n)) respectively (here λ(n)
are the coefficients of F1).
The “cosine” analog of (1.23) holds. To single out F+, we impose the condition
that λ(n) = 0 for χ(n) = −1. We reduce the size of the “cosine” system
(1.23) accordingly, and set λ(1) = 1, and use our two parallel Y. This process
determines λ (the eigenvalue) and F+. If we instead impose the condition
λ(n) = 0 for χ(n) = 1, and set λ(p0) = 1 where χ(p0) = −1 and use the appropriately
reduced “cosine” system with parallel Y , we obtain aF− with some
nonzero a ∈ R. To get F1 and F2, we form 2F+ ± 2i
1
a aF− and determine a
either by Prop. 1.2.4(a) or by use of relation (1.12). If desired, λ can again be
solved for at this stage of the game.
10) An alternative reduction in the size of the “cosine” system (1.23) can be
achieved by looking at the action of WQ on M(Γ0(N), χ,λ). Cf. Prop. 1.2.7
and the discussion afterwards. (This discussion can be generalized to (n,N) =
1 by induction; notice too that there are no oldforms when χ is primitive.)
Each WQ (Fj) must be a nonzero multiple of some Fk by virtue of its Hecke
action. Depending on whether m = 1 or 2, the resulting permutation structure
must either be (1), (12), or (1)(2). In each instance, the relation W 2 Q = ±Id
affords one some a priori control on any “non zero constants” that arise.
Let {WQ1 ,...,WQr } be any commutative family of WQ’s satisfying W 2 Q = Id
and the hypotheses of Prop. 1.2.5. Cf. [6, prop. 1.4]. Since each WQ acts
unitarily, it is natural to seek a basis of MR(Γ0(N), χ,λ) consisting of simultaneous
eigenfunctions of the WQ j . By Prop. 1.2.6, the Fourier expansion of
any such basis function at the cusp Q j
N will be “redundant”; there is thus a
corresponding reduction in the size of the “cosine” analog of (1.23).
Remark 1.3.2. If Cor. 1 applies, the reduction with respect to the WQ’s, as
mentioned above, can be applied also to the case of the trivial character in a
similar manner.
Remark 1.3.3. Note that in the case of N = 4 and trivial character, using
Thm. 1.2 and the fact that the cusp normalizing map of the cusp p3 = − 1
2 ,
σ3 : z ↦→ z
2z+1 is in the normalizer of Γ0(4) we can actually use pure sine/cosine
series and real coefficients at all cusps in this case too. And we can also use
reduction with respect to σ2 = ωN and σ3. This is not true in general for nonsquare-free
levels, e.g. Γ0(9). Then we must use complex arithmetic.
39

Example 1.3.1. As an illustration of (B) consider the case of prime level N ≡
1 mod 4 and χ =
N . With the notation as in (B), for the sake of simplicity,
·
suppose that m = 2 and ε1 = ε2 = 1. Let F1 and F2 = F1 be the Hecke eigenbasis
of normalized newforms in M(Γ0(N), χ,λ). There is only one WQ, i.e. ωN,
which is now also a cusp normalizing map, and ω2 N = Id. We will see how to
diagonalize with respect to ωN to get a reduction of the linear system. Observe
that Tn (ωNF1) = χ(n)ωNTnF1 = χ(n)λ(n)ωNF1, and hence by multiplicity one
we must either have ωNF1 = ±F1 (since ω2 N = Id) or we have ωNF1 = µF2 and
ωNF2 = µ −1F1, for some constant µ ∈ S1 (again since ω2 N = Id). In case
that ωNF1 = ±F1 we already have an eigenfunction of ωN in F1 and we don’t
have to do anything more. If µ = ±1 then the functions F±(defined under (B))
are also invariant under ωN, i.e. ωNF± = ±F± if µ = 1 and ωNF± = ∓F± if
µ = −1. If µ = ±1, then we can form
f± = 1
1 ± µ (F1 ± µF2).
It is easily verified that f± are both unit normalized (have first Fourier coefficient
equal to 1), real-valued on H, and satisfy ωN f± = ± f±. Denote
the Fourier coefficients of f+ and f− by c + (n) and c − (n), respectively. For
(n,N) = 1, one immediately checks that
and
c + (n) = c − (n) = λ(n) for χ(n) = +1
c + (n)c − (n) = λ(n) 2 ≤ 0 for χ(n) = −1.
To find f+, we use the “cosine” analog of (1.23). We reduce the size of the
system by using the fact that ωN is a cusp normalizing map, i.e. we set
c2(k) = c1(k), k = 1,...,M0
together with c1(1) = 1. And then we drop the corresponding equations. Similarly
for f−. In this case, when we run with the parallel Y, we have a lot of
extra tests; Hecke relations coming from the λ(n), c + (n) = c − (n) if χ(n) = 1
and c + (n)c − (n) ≤ 0 if χ(n) = −1. All these can be used as independent tests
for the accuracy of our program.
By running through the search for λ both for f+ and f− we get further
insurance.
Locating the Eigenvalues
We will see how, with the use of the normalized linear system introduced
above, we can locate an eigenvalue R. Cf. [44, p. 5]
Consider the linear system (1.23). The entries of the coefficient matrix
40

V = V (R,Y ) are clearly real-analytic functions of R, but as functions of Y
they might be only piecewise continuous. For a value of R corresponding to
a Maass waveform f we expect the solution vector C = C(R,Y ) to give us
the actual Fourier coefficients if the appropriate normalization is used (e.g.
c(1) = 1), and Y is kept less than Y0 for the sake of well-conditioning.
This fact is used to determine whether a given R is close to a true eigenvalue
or not by solving for C(R,Y ) and looking for known properties of Fourier
coefficients of a Maass waveform. The three most important properties (here)
are:
a) Invariance under change of Y . As described above, for a true
eigenvalue the solution vector will be invariant when we change
Y (keeping Y < Y0).
b) Hecke relations. We can look for Hecke eigenfunctions by seeking
solutions which satisfy multiplicative relations (cf. (1.12)),
for example the relation c1(2)c1(3) = c1(6).
c) Involutions of (Γ0(N), χ). We use cusp normalizing maps which
are involutions with eigenvalues, µ j = ±1, and seek solution vectors
which satisfy c j(k) = µ jc1(k) (cf. (1.14), Example 1.3.1, and
B item 10).
For a given interval I = [R1,R2], we want to find all Maass waveforms with
eigenvalues R in this interval. The idea here is to do this by solving the linear
system (1.23) for two different values of Y in parallel and then try to find R ∈ I
such that the solution vectors C(R,Yi) match (usually we only require the first
few coefficients to match, and use the rest only as an insurance).
In practice we first divide I into a number of equally sized small chunks:
R1 = x0 < ··· < x j < ··· < xN1 = R2,
where the number N1 is chosen in a way that it is probable that any interval
Ij = [x j,xj+1] contains at most one eigenvalue, using e.g. Weyl’s law (1.3) and
some assumption on the behaviour of the nearest-neighbour spacings (e.g. that
they are exponentially distributed).
We then look at each small Ij to see if there is a change of sign in any
of the differences ck − c ′ k , k = 2,3,4, where ck = ck(R,Y1) and c ′ k = c′ k (R,Y2)
are entries in C(R,Y1) and C(R,Y2) respectively. If there are sign changes
for all considered differences in an interval Ij0 we go to the next stage of the
investigation and “zoom in” into this interval. At this point we usually form a
functional
h(R) = ω2(c2 − c ′ 2) + ω3(c3 − c ′ 3) + ω4(c4 − c ′ 4),
41

and try to minimize it over this interval. We use ω j ∈ {±1} to “align” the
differences so that h(R) changes sign where all three differences change sign.
The minimization can be done in a number of ways. One very efficient approach
is to use the method of false position to get successively better approximations
to the location of the minimum (which if it exists is near a point
where h changes sign).
When a value of R that approximates a zero of h is found it is listed as a
candidate for a true eigenvalue. These candidates are stored and subjected to
further examination.
Remark 1.3.4. Zeros of the K-Bessel functions in the left hand side of (1.20)
(or “random noise”) can trigger false indications of zeros in the intervals Ij0 .
Keeping track of how fast c j − c ′ j changes over each interval, most of these
“false” intervals are immediately discarded.
If we only use two Y -values and a few coefficients, it can not be excluded that
the located minimum of h is not close to a real eigenvalue, but such mistakes
will be spotted either when we try to refine the eigenvalue or when we look
closer at properties a)-c) for a larger set of coefficients.
Eventually we declare that the R which we have found is close to an eigenvalue
of the Laplacian, and that our {c j(n)} are close to the Fourier coefficients
of the corresponding Maass waveform. It is worth stressing that, strictly
speaking, these assertions are never proved rigorously through our computations
— although the excellent agreement seen when testing properties a) -
c) on a large number 1 of coefficients on top of those used in the functional
h(R) clearly gives a very strong heuristic justification. The question of giving
rigorous proofs is dealt with in a forthcoming paper [15].
Remark 1.3.5. To speed up the process of looking for sign changes over the
large number of small intervals Ij we use Lagrange interpolation to evaluate
the matrix V at this point (usually interpolation of degree 14 is good enough).
Cf. [44, p. 6].
It should be remarked that the algorithm reproduces both the known oldforms
from Γ0(1) (see [45] or [42]), the newforms on Γ0(5) as in [9], as well
as the explicitly known CM-forms (cf. [46]).
There is obviously no guarantee that we find all eigenvalues in a specified
interval in this manner, but comparing the results with the detailed version of
Weyl’s law (1.3) might give us an indication of missing eigenvalues (cf. [9,
§A.1] and [8, §8]).
1 e.g. 50-10000 produced by the Phase 2 algorithm, cf. Section 1.3.3
42

The Pullback Algorithm
Since the implicit automorphy (1.19) plays an important part in the algorithm,
it is crucial to have an efficient means to compute the pullback z ∗ ∈ FN of a
point z ∈ FN (observe that the notation z ∗ differs from the one used earlier).
We recall that in the case of the modular group Γ0(1), it is easy to make a
pullback to the standard fundamental domain, F1 = {z = x + iy ∈ H||x| ≤
1
1
2 , |z| ≥ 1} using a sequence of “flip-flops” through the generators E : z ↦→ − z
and S : z ↦→ z + 1 (cf. [84, p. 51] or [9, pp. 44-46]).
Instead of extending this algorithm to the case of Γ0(N) by using side pairing
generators of a suitable fundamental domain, we will use the facts that
Γ0(N) is a subgroup of finite index in PSL(2,Z), and that it is easy to find a
set of coset representatives.
Let {Vj} vN
j=1 and FN = ∪Vj(F1) be as on p. 14. Given z ∈ FN we make a
pullback into F1; ˜z = T (z) ∈ F1 with T ∈ Γ0(1). Then we find the index j such
that T −1 ∈ Γ0(N)Vj, and note that VjT ∈ Γ0(N). Hence the Γ0(N) pullback of
z is given by
z ∗ = VjT (z) ∈ FN.
This gives a pullback algorithm for any N, but we need the coset representatives
{Vj} vN
vN
j=1 to apply it. The observation that {Vj} j=1 is by definition a
maximal set of vN = N ∏p|N 1 + p−1 maps in PSL(2,Z), all independent
over Γ0(N) allows us to use the following simple recursive algorithm:
"Traverse Γ0(1) as a tree in S, E and S −1 and collect maps independent over
Γ0(N) until exactly vN independent maps have been found."
This calculation is done once and for all for each group and the resulting representatives
are stored. A possible alternative here is to use explicit formulas
for the coset representatives, but the advantage of the above algorithm is that
it can be extended to other congruence subgroups, such as Γ1(N), Γ 0 (N) or
Γ(N) as well. Actually it can be extended to any finite-index subgroup of
PSL(2,Z) that possesses some precise characteristic which we can use to see
if a given map belongs to the group or not.
1.3.3 Phase 2
After Phase 1 is completed we have an (approximate) eigenvalue R and a
corresponding set of (approximate) Fourier coefficients
c j(k)
1 ≤ j ≤ κ, 1 ≤ |k| ≤ M0 .
43

Suppose now that we want to compute c j(n) for NA ≤ n ≤ NB. Going back to
the identity (1.20),
c j(n)κn(Y ) =
κ
∑
∑
i=1 |k|≤M0
ci(k)V ji
nk + 2[[ε]]
(valid for Y < Y0, 1 ≤ |n| ≤ M(Y ) < Q and 1 ≤ j ≤ κ), we see that in order
to compute c j(n) for n ∈ [NA,NB] (where NB > M0), we take Y < Y0 such that
M(Y ) > NB. To avoid cancellation we also have to make sure that the particular
choice of Y does not make κn(Y ) too small. We can now formulate the core
of Phase 2 as a theorem.
Theorem 1.7. Suppose that f ∈ M(Γ0(N), χ, 1
4 + R2 ) has Fourier coefficients
c j(n) . Let Y < Y0, and let n,Q be integers such that Q > M(Y ) and 1 ≤
|n| ≤ M(Y ). We then have
c j(n) = ∑κ ji
i=1 ∑|k|≤M0 ci(k)Vnk + Errn(Y ), (1.24)
κn(Y )
where V ji
nk is given by (1.21) on p. 36, and the error Errn(Y ) is given by
Errn(Y ) = [[ε]]
κn(Y ) .
In order to use Theorem 1.7 to successfully compute the Fourier coefficients
we have to adjust the value of Y (and accordingly M(Y ) and Q also) to keep
the error Errn(Y ) from growing.
If we just want to confirm the existence of an eigenvalue in the neighbourhood
of R, or if we want to make a picture of an eigenfunction, the first set of
coefficients from Phase 1 is usually sufficient. But, even for those purposes,
the method of Phase 2 is a very cheap way to improve the last part of the
already obtained coefficients, as well as obtaining a lot more.
1.3.4 Remarks on the Performance of the Algorithm
The main difference (with regard to performance issues) between groups with
one cusp, e.g. the modular group or Hecke triangle groups, and the groups
Γ0(N) with N > 1 should now be clear. The presence of extra cusps introduces
more sets of coefficients, thus increasing the size of the linear system used in
Phase 1. We will give some examples of how the relevant factors scale with
respect to N (and R).
44

Table 1.1: Time to compute compute the coefficient vector for R = 9.5336... on a
3.2GHz CPU
Timing for a Single R
N M0 T (R,N)
2 13 0.03s
5 32 0.14s
13 90 1.17s
17 118 2.20s
41 307 23.9s
101 757 287.4s
Suppose we are given a level N and a potential eigenvalue R, and that we want
to compute the corresponding (minimal) set of coefficients, C = {c j(n)|1 ≤
|n| ≤ M0, 1 ≤ j ≤ κ}, i.e. we have to solve the system (1.22) under some normalization
assumption. Let T (R,N) be the time required to perform this task.
It turns out in practice that the time required to solve the system is negligible
compared to the time required to compute the coefficient matrix, and the
main contribution to T (R,N) in fact comes from the K-Bessel computations.
Observe that in the evaluation of V ji
nk the only K-Bessel terms we need are
) for 1 − Q ≤ m ≤ Q, these are independent of n and i, hence the
KiR(2π|k|y ∗ m j
total number of K-Bessel computations is 2Q(M0κ) ∼ 2κM2 0 (as we can take
Q ∼ M0). Using Y0 ≥
√ 3
2N we see that, roughly M0 = [const]N(R + 1), and
T (R,N) ≪ κN 2 (R + 1) 2 . (1.25)
Remember that κ is dependent on the number of prime factors of N, and for
a prime number κ = 2. From this formula we see that the time to compute
even a single eigenfunction with a small eigenvalue increases drastically as
the level N grows. No matter how much we improve the speed of computing
the individual K-Bessel functions, attaining levels like N ≈ 1000 seems to
be out of reach at the present time. In table 1.1 we give some examples of
T (R,N) for different (prime) N, with R = 9.5336... (the first eigenvalue on
PSL(2,Z)).
Recall steps 1-3 on p. 41. To find eigenvalues we need to compute the
matrix V (R,Y ) at least 14 × 2 = 28 times for step 1 (using Lagrange interpolation
of degree 14 for the function R ↦→ V (R,Y ), cf. Remark 1.3.5) and then
at least 6×2 = 12 times for the step 3 (assuming that it takes approximately 6
iterations with the method of false position to obtain the required precision).
45

The K-Bessel Function
The K-Bessel function is computed using a number of different algorithms,
each with different range of applicability (cf. [42], [98] and [9]).
One possible way to speed things up is to use an adaptive Lagrange interpolation
method for the function x ↦→ KiR(x). However, due to the oscillatory
nature of KiR(x) for small x we need a large number of interpolation points
when N is large (i.e. when Y0 is small) so this type of interpolation is in general
only efficient in Phase 2 or when R is large .
Timing for Phase 2
Since we only use the first few V ji
nk with |k| ≤ M0 in Theorem 1.7 we need
about 2QM0κ K-Bessel calls to compute c j(n). Remember here that for (1.24)
to hold we need Q > M(Y ) > |n|, and we also need Y < const (1 + R)/n,
hence, if we take Y = [const] · (1 + R)/n,then we have M(Y ) = [const] · (1 +
R)/Y = [const] · n. The time to compute one coefficient c j(n) is thus roughly
proportional to
nκM0 = κN(R + 1)n,
and hence the time to compute the first A coefficients is approximately proportional
to
κN(R + 1)A 2 , as A → ∞.
Observe that the K-Bessel functions in V ji
nk does not depend on n or i. Hence
to avoid unnecessary computations we can use a vector to store all 4QκM0
K-Bessel values used in (1.24). We will then only need to perform these
computations when we change Y, but unfortunately we will have to deal with
memory issues instead, and these can degrade the performance as much as
the computations themselves. To compute a large number of coefficients we
have to keep a balance between the number of computations and the size of
the allocated memory. In practice this is a rather difficult task.
1.4 Results
The focus of the current project has been to provide a robust and efficient
algorithm for computing Maass waveforms on Hecke congruence subgroups
Γ0(N) for any integer N. The following are our main results so far.
1. Extensive lists of eigenvalues for small and prime N = 5,7,13, with either
trivial
character or the real Dirichlet character mod N given by the symbol
N
· .
2. Shorter lists of eigenvalues for all N ≤ 30. More precisely, we believe our
lists contain all eigenvalues with R ≤ 10 for N = 1,2,3,4,5,6,7,8,9,10, 11,
46

12, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26,29, R ≤ 5 for N = 16,20,24, R ≤ 3
for N = 30 and R ≤ 2 for N = 18,27 and 28.
3. The first eigenvalue, λ1 = 1
4 + R21 , for all prime N ≤ 131.
4. Large sets of Fourier coefficients for a few Maass waveforms, e.g. for N =
5 the following coefficients have been computed: all c(n) with |n| ≤ 70209
for R = 3.2642...; all c(n) with |n| ≤ 89263 for R = 4.8937...; all c(n)
with |n| ≤ 56420 for R = 4.1032... (the first two eigenvalues correspond
to the character
N
· and the last one corresponds to the trivial character).
Work on extending these data, as well as carrying out more extensive statistical
tests, is currently being pursued. In this section we will give some
examples and comments on each item in the above list.
Further data than that which is presented in this section is available from
the author upon request.
1.4.1 Eigenvalues
Tables 1.6, 1.3 and 1.4 provide some examples of eigenvalues for different
groups. In tables 1.3 and 1.4 the eigenvalues of the operators J and ωN are
indicated in the respective columns. The eigenvalues are denoted by − for
−1, + for 1, and ∗ is used to indicate that both eigenvalues are present. In all
tables we use H1 = |c1(2)c1(3) − c1(6)| as one indication of the accuracy of
the program (we based our search on property a) on p. 41; hence no Hecke
relation like H1 was built in to our algorithm).
It should be remarked that in the case of Table 1.6, Γ0(4), all computed
newforms are eigenfunctions with eigenvalue −1 of both non-trivial cusp normalizers,
i.e. σ2 = ω4 : z ↦→ −1/4z, and σ3 : z ↦→ z/(2z + 1), which are both
Γ0(4)−involutions.
1.4.2 Lowest Eigenvalues
We have used the Phase 1 algorithm to compute small eigenvalues for the
groups M(Γ0(N), χ) as N ranges through the primes up to 131 and χ is either
trivial or the real Dirichlet character
N
· . The first located eigenvalues are
listed in Table 1.2.
Recall that all eigenvalues fall into distinct classes with respect to the involutions
J and ωN. We denote the classes by {++,+−,−+,−−}.
In the case of a non-trivial character, CM-forms are present, and in all cases
considered such a form was found to occur as the lowest eigenvalue. All the
CM-forms occurring here are of the type considered in [20] (i.e. the narrow
class number of Q[ √ N] is 1), and by the explicit formula (cf. [20, p. 112]) the
CM-forms with lowest eigenvalues are of the type − − . In table 1.2, H1and
H2 are two parameters indicating the error. H1 is always defined by H1 =
47

Figure 1.1: Histogram of Fourier coefficients for Maass forms on Γ0(5), 5 ·
350
300
250
200
150
100
50
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
(a) R = 3.2642... (CM-form)
250
200
150
100
50
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
(b) R = 4.8937...
|c1(2)c1(3) − c1(6)|, and H2 is defined as either the true error in R for CM-
forms, or as H2 = |c1(N)| − 1
(cf. Prop. 1.2.4) otherwise.
√ N
1.4.3 Fourier Coefficients
Table 1.5 gives some examples of Fourier coefficients and Hecke relations
obtained by the Phase 2 algorithm. Figure 1.1 (a) is a histogram of the co-
efficients c1(p) of the CM-form with eigenvalue 3.26... where the first 3407
primes for which χ(p) = = 1 are used. Figure 1.1 (b) is a histogram
5
p
of the ˜c1(p) for one of the Maass waveforms on (Γ0(N), χ5) with eigen-
5
value R = 4.8937.... Here ˜c1(p) = c1(p) if χ(p) = p = 1 and −ic1(p)
5
if χ(p) = p = −1 (to get real coefficients), and the first 7462 primes are
used. The experimentally obtained density functions (curves) in Figures 1.1
(a) and (b) should be compared with the known density function 1
π 4 − x2 − 1 2
in the CM-case (a) and the conjectured density function 1
√
2π 4 − x2 in the non-
CM case (b) (cf. [46, 45]).
48

Table 1.2: Lowest eigenvalues for Γ0(N)
N χ J ωN R H1 H2
2 - - 5.4173348068447 0.3E-13 0.5E-12
3 - - 4.38805356322 0.1E-10 0.8E-11
5 - - 3.02837629306 0.4E-11 0.7E-12
5· 5 - - 3.264251302636 0.7E-11 0.2E-12 CM
7 - - 1.92464430511 0.2E-11 0.8E-11
11 - - 2.03309099385 0.3E-12 0.1E-11
13 - - 0.9708154174 0.7E-10 0.2E-10
13· 13 - - 1.31473442107 0.1E-10 0.7E-12 CM
17 - - 1.4414285450 0.5E-10 0.3E-10
17· 17 - - 0.7498863406 0.5E-10 0.4E-10 CM
19 - - 1.0919915598 0.2E-13 0.3E-10
23 - + 1.3933371415 0.6E-10 0.2E-10
29 + + 1.017266551 0.5E-09 0.8E-09
29· 29 - - 0.953598000 0.3E-11 0.5E-10 CM
31 + + 0.789356178 0.8E-10 0.4E-08
37 - - 0.64230596 0.4E-08 0.6E-08
37· 37 - - 0.630391294 0.7E-09 0.7E-09 CM
41 + + 0.66572483 0.9E-08 0.3E-08
41· 41 - - 0.37767450 0.2E-08 0.1E-07 CM
43 - - 0.65545239 0.5E-09 0.7E-08
47 - + 0.5854522 0.5E-07 0.1E-08
53 - - 0.8039894582 0.2E-12 0.4E-08
53· 53 - - 0.799094454 0.5E-10 0.8E-09 CM
59 + + 0.595829688 0.4E-10 0.3E-08
61 - - 0.4180624 0.4E-07 0.9E-07
61· 61 - - 0.42868522 0.2E-08 0.3E-08 CM
67 - - 0.67759021 0.3E-07 0.1E-08
71 + + 0.35745048 0.8E-08 0.5E-08
73 - - 0.517887505 0.1E-08 0.2E-09
CM This is a CM-form. For definition of H2 see the end of the table
49

N χ J ωN R H1 H2
73·
73
- - 0.20488585 0.5E-07 0.7E-08 CM
79 + + 0.55177485 0.2E-06 0.9E-12
83 - + 0.640287578 0.3E-09 0.3E-09
89 + + 0.4894360 0.1E-07 0.1E-07
89· 89
- - 0.2273961 0.3E-08 0.6E-07 CM
97 - - 0.41693167 0.1E-07 0.8E-08
97· 97
- - 0.16846 0.4E-05 0.1E-05 CM
101 + + 0.45375925 0.6E-09 0.3E-08
103 - - 0.56540670 0.9E-08 0.5E-10
107 - + 0.5816789 0.2E-07 0.8E-07
109 - - 0.423655 0.2E-05 0.9E-07
109 - - 0.2822870 0.8E-09 0.3E-07 CM
109
·
113 - - 0.46200265 0.4E-09 0.3E-07
113 - - 0.21379233 0.2E-08 0.8E-08 CM
113
·
127 + + 0.373385109 0.5E-07 0.4E-10
131 - + 0.261072758 0.7E-09 0.4E-11
CM This is a CM-form
For a CM-form H2 is the true error and for a non-CM form we know
by Proposition 1.2.4(a) that |c(N)| = 1, and we put H2 = ||c(N)| − 1|.
Table 1.3: Eigenvalues for N = 29, χ =
29
·
50
R J ωN H1 H2 a
0.9535979998 - - 0.9E-10 0.5E-10 (CM)
1.804894697229 - * 0.2E-13 0.2E-12
1.907196000095 + + 0.3E-13 0.6E-12 (CM)
1.9491483792 + * 0.9E-14 0.2E-10
2.1682948312 - * 0.7E-13 0.3E-10
2.8084567797 - * 0.8E-12 0.4e-10
2.8252375274 + * 0.2E-13 0.8E-10
2.8607940001 - - 0.5E-13 0.1E-11 (CM)
a H2 is the true error in R for CM-forms and otherwise
|c+(4) − c−(4)|. Cf. Remark 1.3.1.

Table 1.4: Eigenvalues for N = 23, χ = 1
R J ωN H1 H2 = ||c(23)| − 1
√ 23 | a
1.3933371415 - + 0.8E-11 0.4E-11
1.5061266371 - - 0.3E-11 0.1E-10
1.5795892401 + + 0.7E-12 0.3E-11
1.9352504926 - - 0.3E-11 0.3E-10
2.4307774166 - - 0.1E-10 0.9E-10
2.4643338705 + + 0.7E-11 0.9E-09
2.4936237868 - + 0.2E-11 0.7E-11
2.5321343295 + + 0.1E-11 0.1E-09
2.6109599620 + - 0.4E-12 0.3E-11
2.7362777132 - + 0.1E-10 0.1E-09
2.8079149467 - - 0.6E-13 0.1E-10
3.0166209534 + - 0.3E-11 0.5E-11
3.2357419725 + + 0.2E-12 0.2E-10
3.3723339187 - - 0.2E-10 0.8E-11
3.5202562438 + - 0.1E-11 0.8E-13
3.5245436383 - + 0.5E-10 0.1E-09
3.5882850865 - + 0.9E-12 0.3E-11
3.7062705391 - - 0.6E-10 0.2E-12
3.8167552363 + - 0.3E-12 0.4E-12
3.8189248762 - - 0.5E-10 0.2E-09
3.8400352089 + + 0.6E-11 0.1E-11
3.9122175318 + + 0.9E-10 0.7E-10
4.16025275662 + + 0.5E-12 0.1E-11
4.19680364676 - - 0.6E-11 0.4E-11
4.23167066214 - + 0.9E-10 0.5E-09
4.30661013365 - - 0.7E-11 0.8E-11
4.43672242684 + + 0.4E-10 0.8E-08
4.44033811150 + + 0.7E-10 0.6E-08
4.50005039136 - + 0.6E-10 0.4E-11
4.50100759650 + - 0.1E-11 0.2E-12
4.73923930491 - + 0.1E-11 0.5E-12
4.78525397008 - - 0.7E-12 0.3E-11
4.83751695193 - - 0.5E-11 0.6E-11
4.85889850005 + - 0.2E-10 0.8E-10
4.88226240518 + - 0.3E-10 0.1E-10
4.98487107442 - + 0.4E-10 0.1E-08
4.99908300143 - + 0.2E-10 0.6E-10
a From Proposition 1.2.4(b) we know that |c(23)| = 1
√23 .
51

Table 1.5: Fourier coefficients for N = 5 and χ =
5
·
52
R = 4.893781291438 R = 3.264251302636 (CM) a
c(2) 1.217161411799i
c(3) 0.295119713347i
c(5) b exp(1.157414657530i)
c(7) −1.138873755146i
c(11) −0.041396292578
c(13) −0.558344591841i
c(17) −0.212576664102i
c(19) 0.608670097807
c(23) −1.205831908853i
c(29) 0.162328579004
c(31) −0.556019364974
c(37) 0.411889174623i
c(41) −0.835153489179
c(43) −1.240299728133i
c(47) −0.605458209042i
c(53) 1.458187614411i
c(59) 0.922374897976
c(61) −1.247703551349
c(67) 0.33772433772i
c(71) −0.74219625846
c(73) −0.44062515256i
c(79) 0.71467631335
c(83) 0.99829038428i
c(1553) −1.3885884029i
c(72977) −0.9912088597i
c(2) 0 0.1E-12
c(3) 0 0.1E-12
c(5) -1 0.1E-12
c(7) 0 0.1E-12
c(11) −1.11318397521973 0.2E-07
c(13) 0 0.4E-11
c(17) 0 0.1E-10
c(19) −0.17157173156738 0.6E-08
c(23) 0 0.5E-09
c(29) 0.44610100984573 0.1E-07
c(31) −0.16668809652328 0.6E-08
c(37) 0 0.4E-11
c(41) 0.84569376707077 0.1E-07
c(43) 0 0.1E-10
c(47) 0 0.6E-10
c(53) 0 0.1E-09
c(59) −0.59840279817581 0.5E-08
c(61) −1.82808148860931 0.4E-07
c(67) 0 0.1E-08
c(71) 1.29922688007355 0.5E-07
c(73) 0 0.1E-12
c(79) −1.48637688159943 0.2E-07
c(83) 0 0.1E-12
c(89) 1.43317067623138 0.5E-07
c(97) 0 0.1E-12
Some Corresponding Hecke relations (for R = 4.893781291438)
|c(2)c(3) − c(6)| 0.2E-12
|c(93) − c(3)c(31)| 0.3E-11
|c(895) − c(5)c(179)| 0.1E-11
|c(15625) − c(5) 6 | 0.1E-10
||c(5)| − 1| 0.1E-12
|c(195) − c(3)c(5)c(13)| 0.2E-11
|c(9763) − c(13)c(751)| 0.3E-10
|c(72991) − c(47)c(1553)| 0.2E-09
a Since coefficients of CM-forms can be computed explicitly (cf. [46, p. 5]), the first
column contains the coefficients computed according to the explicit formula, and
the second column is the difference between this value and the value obtained with
the methods presented in this paper.
b By Proposition 1.2.4(a) we know that |c(5)| = 1.

Table 1.6: Eigenvalues for Γ0(4)
Sine Cosine
R H1 H2 c
3.70330780123 5E-12 7E-12
b 5.41733480684 2E-13 1E-12
6.62042287384 1E-12 5E-13
b 7.22087197596 2E-12 4E-12
b 8.27366588959 4E-12 1E-12
8.52250301688 8E-13 2E-13
a 9.53369526135 3E-12 4E-12
9.93491995937 3E-13 7E-13
b 10.71270690070 2E-13 3E-10
10.76471068319 3E-13 2E-13
b 11.31767970147 4E-13 5E-13
11.97277669404 1E-12 4E-12
a 12.17300832468 2E-12 2E-12
b 12.82198816197 8E-13 4E-13
b 13.31016428347 9E-13 2E-12
b 14.09720373392 9E-12 2E-12
14.1438850658 2E-11 1E-11
a 14.3585095183 2E-11 4E-11
14.97031306857 2E-12 2E-12
b 15.27402248481 2E-12 3E-13
b 15.4428777438 7E-12 6E-11
15.8093206166 2E-11 1E-11
a 16.1380731715 3E-13 2E-11
16.4782589777 1E-11 1E-12
a 16.6442592019 1E-11 8E-12
b 17.3193334036 8E-12 2E-11
b 17.4931127227 2E-11 3E-11
17.6699198653 2E-11 1E-12
R H1 H2 c
5.879354157759 3E-13 6E-13
8.042477591693 7E-12 4E-13
b 8.92287648699 5E-12 2E-12
9.859896162239 7E-13 9E-13
b 10.92039200294 4E-12 3E-12
b 12.09299487508 1E-13 3E-11
12.87761656411 1E-12 2E-13
13.17207496748 5E-12 7E-12
a 13.77975135189 2E-12 1E-10
14.47780273166 1E-13 7E-13
b 14.68501595060 2E-10 7E-12
b 15.31419658418 9E-12 2E-12
15.7434456453 2E-11 6E-11
16.09359237739 7E-12 6E-12
b 16.4041087751 5E-11 7E-11
c 16.94026094636 1E-11 3E-12
17.19512721203 8E-12 2E-12
a 17.73856338106 8E-12 1E-11
b 17.8780025394 2E-11 2E-11
18.24637789203 1E-12 3E-12
18.6207750965 1E-11 2E-11
b 19.12542240816 8E-12 1E-11
a 19.42348147083 9E-12 6E-12
19.69901979586 1E-12 7E-12
19.7785769776 1E-11 7E-11
a Oldform from Γ0(1). This eigenspace is 3-dimensional.
b Oldform from Γ0(2). This eigenspace is 2-dimensional.
c From Proposition 1.2.4(c) we know that |c(4)| = 0 for newforms, and it is easy
to see that in an oldspace f±(z) = f (z) ± ε f (dz), d|4, ε ∈ {±1} (cf. notation
in Remark 1.3.1, and hence c+(n) = c−(n) for (n,4) = 1. Accordingly we set
H2 = |c(4)| for newforms and H2 = |c+(5) − c−(5)| for oldforms.
53

2 Computation of Maass Waveforms with
Non-trivial Multiplier Systems
2.1 Introduction
In this chapter we will consider computational aspects of Maass waveforms
on the modular group PSL(2,Z), and the Hecke-congruence subgroups Γ0(N),
non-zero weight and a non-trivial multiplier system.
First we will introduce the basic theory of multiplier systems and then we
will introduce the notion of Maass waveforms in this context. We will also
provide some details on the different operators that acts on spaces of Maass
waveforms with non-trivial multiplier system.
As we will see in section 2.8, we only need to make a few minor changes in
the algorithm we used to compute Maass waveforms of weight 0 (cf. Chapter
1) to adapt it to the case of non-zero weight and non-trivial multiplier system.
The biggest obstacle to overcome is the numerical evaluation of the Whittaker
function, which is the non-zero weight analogue of the K-Bessel function.
The algorithm used to compute this function is presented in Chapter 4.
The last section contains some examples of the results that has been obtained
with the described methods. To my knowledge, the only previous computational
investigation of Maass waveforms with non-zero weight was performed
by Mühlenbruch, [70], but his method has less accuracy and a smaller
range of applicability than the methods to be presented in this chapter.
2.2 Multiplier Systems
2.2.1 Introduction
We will give a brief introduction to multiplier systems, for more extensive
treatments see [39, pp. 331-338], [77], or [84, pp. 70-87].
Let Γ be a Fuchsian group and m be even. Classically, a function ϕ, meromorphic
on the upper half-plane H, which satisfies
ϕ(Az) = ΘA(z;m)ϕ(z) = (cz + d) m
a b
ϕ(z), ∀A = ∈ Γ, (2.1)
c d
55

is called an automorphic form of weight m. The function
ΘA(z;m) = (cz + d) m
, A =
a
c
b
d
∈ Γ
is said to be an automorphy factor on Γ. The classical theory of automorphic
forms is well-known; for instance, if m = 2, then the automorphic forms can be
identified with the meromorphic differential forms of degree 1 on the orbifold
(classical Riemann surface) Γ\H.
We observe that, for even m, the number (cz + d) m is uniquely defined and
the automorphy factor ΘA(z;m) in (2.1) clearly satisfies
ΘA(Bz;m)ΘB(z;m) = ΘAB(z;m). (*)
To generalize these notions to arbitrary real m, there needs to be a choice
of branch of the argument, and to make certain everything is well-defined, we
have to introduce the notion of a multiplier system.
Definition 2.2.1. We will always use the principal branch of the argument,
−π < Arg(w) ≤ π. Define
jA(z;m) = e imArg(cz+d) m
(cz + d)m cz + d 2 a b
= =
, A = ∈ SL(2,R).
|cz + d| m cz + d
c d
To adapt the relation (*), we also write
σm(A,B) = jA(Bz;m) jB(z;m) jAB(z;m) −1 . (2.2)
It is clear that for integer m, σm(A,B) = 1, but it can also be shown (cf. [77,
§2, pp. 42–50]) that the only values which σm can take are {1,e ±2πim }.
It is customary to write σm(A,B) = exp[2πimw(A,B)] as in [84, eq. (3.2.1)].
Let Γ be the inverse image of Γ under the natural homomorphism SL(2,R) →
PSL(2,R). (Note that this forces −I ∈ Γ.)
Definition 2.2.2. A multiplier system v of weight m on Γ is a function v : Γ →
S 1 (where S 1 ⊂ C is the unit circle), such that
• v(−I) = e −πim , and
• v(AB) = σm(A,B)v(A)v(B), ∀A,B ∈ Γ.
Observe that v can be regarded equally well as a multiplier system of any
weight m ′ ≡ m mod 2. The question of whether there exist multiplier systems
of a given weight and on a given group is most easily answered by the
following proposition (cf. [39, prop. 2.1, p. 333]).
Proposition 2.2.1. Given v : Γ → S 1 and m ∈ R. The following are equivalent:
56

• v(T ) is a multiplier system of weight m on Γ.
• There exists a function ϕ ≡ 0 on H which is either C∞ or meromorphic
such that
ϕ(T z) = v(T )ϕ(z)(cz + d) m
, ∀T =
a
c
b
d
∈ Γ.
A function ϕ as above which satisfies
ϕ(T z) = v(T )ϕ(z)(cz + d) m , ∀T =
a b
c d
∈ Γ,
is called an automorphic form of weight m and multiplier system v on Γ.
Definition 2.2.3. Given a multiplier system v on Γ and an element α ∈ GL(2,R)
we define a multiplier system, vα , on the group α−1Γα by
αAα−1 ,α
v α (A) = v αAα
−1 σm
σm (α,A)
, A ∈ α −1 Γα.
That this indeed gives a multiplier system on α −1 Γα is shown in [63, p.
138].
Using Prop. 2.2.1, we will construct the two most popular multiplier systems
in the following sections. Compare: [39, pp. 334-337].
2.2.2 The η multiplier system
The η-function
The Dedekind η−function is a holomorphic function on H, defined by
z
∞
η(z) = e (1 − e(nz)),
24
∏
n=1
where e(x) = e2πix . Note that η can be expressed by the following Fourier
series:
η(z) =
∑ cne n +
n≥0
1
z ,
24
⎧
⎪⎨ 1, n = 6m
cn =
⎪⎩
2 ± m,
−1, n = 6m2 0,
± 5m + 1,
otherwise.
It is clear from the definition that η(z) = 0 for z ∈ H and that, for each k ∈ R,
η 2k can be defined as a holomorphic function on H. (Cf. [84, p. 205].) It is
57

easily seen that η 2k has a Fourier series of the form
η(z) 2k = e
kz
12
∑ Pn(k)e(nz),
n≥0
where Pn(k) is defined by the relation
j
∏ 1 − q
j≥1
2k = ∑ Pj(k)q
j≥0
j .
Using the expansion 1 − q j2k 2k n
= ∑n≥0 n (−1) q jn , which is valid since
|q| = |e(z)| < 1 for z ∈ H, the first few values of Pj(k) can be evaluated to:
P0(k) = 1, P1(k) = −2k, P2(k) = 2k 2 − 3k, P3(k) = − 4
3 k3 + 6k 2 − 8
3 k,
etc. (see a few more at [70, p. 18]). Note that η(z) 24 is the famous Ramanujan
function ∆(z). Cf. [84, pp. 196-197].
The multiplier system
It can be proved (cf. Thm 3.1 and Thm 3.4 [3, p. 48 and p. 52]) that η satisfies
the following functional equations
−1
η = (−iz)
z
1 2 η (z), and in general,
η(γz) = vη(γ)(cz + d) 1
2 η(z), ∀γ =
a
c
b
d
∈ SL(2,Z). (2.3)
This functional equation expresses the fact that η is a SL(2,Z)−automorphic
form of weight 1
2 and multiplier system given by vη. Accordingly the function
η 2k is an SL(2,Z)-automorphic form of weight k and multiplier system given
by v 2k
η , and we can use η 2k in the context of Proposition 2.2.1 to assure the
existence of the multiplier system, vη,k = v 2k
η , of weight k on SL(2,Z) (and
any of its subgroups, e.g. Γ0(N)).
We have the following explicit formula for v = v2k η :
1
2πi logv
kb
a b
=
12 ,
c d k
a = d = 1,c = 0,
a+d−3c
12c − s(d,c) , c > 0,
(2.4)
and
v
−a
−c
−b
−d
= e kπi
v
a
c
b
d
, for c > 0.
58

Here s(d,c) is the Dedekind sum,
s(d,c) =
c−1
∑
n=1
n
c
dn
,
c
where ((x)) is the saw-tooth function
x − ⌊x⌋ −
((x)) =
1
2 , if x /∈ Z,
0, if x ∈ Z,
and ⌊x⌋ is the greatest integer less than or equal to x. Note that if x is not
an integer, then ⌊−x⌋ = −⌊x⌋ − 1 so ((−x)) = −((x)), and hence s(−d,c) =
−s(d,c) if gcd(d,c) = 1. One can also express s(d,c) as (cf. [3, p. 61])
s(d,c) = ∑
n mod c
n
c
dn
c
.
Remark 2.2.1. It is known that, for each k ∈ R, there exist exactly 6 different
multiplier systems of weight k on PSL(2,Z) (cf. [84, §3.4, pp. 83, 206 ]
or [63, thm. 19, p. 132]). We will denote these by v (r)
η,k
= v2(k+r) η , where r ∈
{0,2,4,6,8,10}. Compare [84, eq. (6.4.7)]; one knows, of course, that v 24
η = 1.
When dealing with the modular group and weight k, it is sufficient to con-
sider only the multiplier system v (0)
later, cf. Section 2.4.5).
2.2.3 The θ Multiplier System
η,k = vη,k = v 2k
η (for reasons to be discussed
On any subgroup of PSL(2,Z), we can always use the η-multiplier system, but
in general, if the subgroup has more generators and relations, there are also
other multiplier systems available. In particular, on Γ0(4), there is a multiplier
system of weight 1
2 which is interesting from an arithmetical point of view.
It is well-known (cf. [90] or [51]) that the Jacobi theta function
θ(z) =
∞
∑
−∞
e(n 2 z), z ∈ H,
is automorphic on Γ0(4) with weight k = 1
2 and can be used to define a multiplier
system on Γ0(4). Using the Poisson summation formula one can prove
(cf. [34, pp. 72–75] or [51, pp. 167–168]) that the θ-function satisfies:
−1
θ = (−iz)
2z
1
z
2 θ , (2.5)
2
59

and one can also prove that (cf. [51, thm. 10.10, p. 177] or [90, p. 447]):
θ(Az) = vθ (cz + d) 1
2 θ(z), A =
a
c
b
d
∈ Γ0(4), (2.6)
where the multiplier vθ can be expressed explicitly as
c
vθ = ¯εd ,
d
where εd = 1 if d ≡ 1 mod 4 and εd = i if d ≡ −1 mod 4, and
c
d denotes the
extended quadratic residue symbol defined as the traditional Jacobi symbol if
0 < d ≡ 1 mod 2 and extended by
c
=
d
c
c
, c = 0,
|c| −d
and
0 1 if d = ±1,
=
d 0 otherwise.
For the sake of completeness we also use the traditional Kronecker extension,
i.e. we define c
2
=
2
c
. (2.7)
Observe that since vθ (−I) = e−πik (which follows from the choice of argument,
−π < Arg(z) ≤ π), we can always suppose that c > 0 when we evaluate
the symbol
c
d .
One can verify that the symbol
·
· defined as above satisfies reciprocity
relations similar to the usual ones:
Proposition 2.2.2. Suppose that c,d ∈ Z are odd and c = 0. Then we have:
−1
d
c
d
d−1
= (−1)
( 2 )
,
=
d
c (−1)
( d−1
2 )( c−1
2 )
, d, or c > 0,
− d
c
Worth noting here is that for odd c,d we have
60
d−1
(−1)
( 2 )( c−1
2 )
=
(−1) ( d+1
2 )( c+1
2 ) , d, and c < 0.
−1, c ≡ d ≡ 3 mod 4,
1, otherwise,

and
d+1
(−1)
( 2 )( c+1
2 )
=
−1, c ≡ d ≡ 1 mod 4,
1, otherwise.
Remark 2.2.2. Relations (2.5) and (2.6) can also be proved using the corresponding
relations (2.3) for η and the following relation between the η and
the θ functions (cf. [51, p. 177] or [55, thm. 12, p. 46])
θ(z) = η2 z+1
2
η(z + 1)
. (2.8)
Remark 2.2.3. By (2.8) it is clear that we can define the function θ(z) 2k
uniquely in H for all k ∈ R, and hence we can also define the multiplier system
v 2k
θ of weight k on Γ0(4). In this chapter, however, we will consider only half
integral weights for the theta multiplier system.
2.2.4 Further properties of the multiplier systems
We are going to consider the eta multiplier system on subgroups of the modular
group. It is possible to express the eta multiplier system without involving
a Dedekind sum, i.e. we have the following formula from Knopp [55, p. 51]:
a b
c
1 2
vη
= e (a + d)c − bd c − 1 + 3d − 3 − 3cd
c d d 24
,
c
d
for c > 0, even. (Note that Knopp’s symbol
c
d
if c = 0).
∗
is equal to our symbol
Remark 2.2.4. Using this formula and the corresponding formula for vθ ([55,
p. 51, thm. 3]) it is easy to see that the eta and theta multipliers coincide on the
principal congruence subgroup Γ(24). Note that [Knopp’s θ](z) =[our θ]
z
2 ,
and that
b
a b
a
[Knopp’s vθ ]
= [our vθ ] 2 , when
c d
2c d
a b
1 0
≡
mod 2.
c d
0 1
Another property of the eta multiplier system is that for integral weights it
reduces to a Dirichlet character on certain subgroups.
61

Proposition 2.2.3. The eta multiplier system at weight 1, v2 η, agrees with the
odd Dirichlet character d−1
−1
d = (−1) 2 modulo 4 on the group
Γ12 = A ∈ Γ(1)|A ≡
∗
0
0
∗
mod 12 ⊂ Γ0(12)
(and hence it is trivial on Γ(24)).
Proof. Suppose that
A =
a 12b
12c d
∈ Γ12.
Then, for c > 0 we have (note that (12,d) = 1)
v 2 η(A) =
1 2
e (a + d)12c − 12bd 144c − 1 + 3d − 3 − 36cd
12
=
1
d − 1
e [3d − 3] = e = i
12 4
d−1 = (−1) d−1
=
2
−1
.
d
If c = 0 we use (2.4) and see that if a = d = 1 then
and if a = d = −1 then
Hence, in all cases, if c ≥ 0 we have
2.3 Maass Waveforms
v 2 η(A) = 1,
v 2 η(A) = −1.
v 2 η(A) =
−1
.
d
The slash-operator f |A(z) = f (Az) can be extended to an operator of weight k
as:
f |[A,k](z) = f (Az) jA(z;k) −1 ,
62

and the natural analog of the Laplace-Beltrami operator, ∆, which is invariant
under this action is the weight-k Laplacian:
∆k = ∆ − iyk ∂
∂ 2 ∂ 2
= y2 +
∂x ∂x2 ∂y2
− iyk ∂
∂x .
If Γ is a Fuchsian group we define the space M(Γ,v,k,λ) consisting of Maass
waveforms on Γ, of weight k, multiplier system v and eigenvalue λ, as the
space of functions which satisfy the following conditions:
1) f |[A,k](z) = v(A) f (z), ∀A =
2) ∆k f + λ f = 0, and
3)
F | f |2 dµ < ∞.
Observe that condition 1) is equivalent to
1’) f (Az) = v(A) jA(z;k) f (z), ∀A =
a b
c d
a b
c d
∈ Γ,
∈ Γ.
For purposes of the computational work to be described in this chapter, we
shall be content to restrict ourselves to cases where λ > 1
4 . (Cf. also here para.
4 of sect. 2.9.1 below.)
Instead of the Bessel equation in the case of weight 0, condition 2) above
gives us the Whittaker equation, and using the method of separation of variables
gives us Whittaker functions instead of the K-Bessel functions at weight
0 (for complete details see [39, chap. 9]). Since f (x+iy) is no longer periodic
in x, but instead satisfies f (z + 1) = v(S) f (z) = e(α) f (z), with α ∈ [0,1), the
Fourier series of f can ([39, pp. 26, 348, 420(19)]) be written as
f (z) =
∞
∑
−∞
n+α=0
c(n)
W k sgn(n+α)
|n + α| 2 ,iR (4π|n + α|y)e((n + α)x), (2.9)
where Wl,µ(x) is the Whittaker function in standard notation (cf. [29, vol. I, p.
264]) and R is the usual spectral parameter, λ = 1
4 + R2 . One notes here that
Wl,µ(x) = e− x 2 x (µ+ 1
2) 1
x
Ψ µ + .
2 − l;2µ + 1;x and that W0,µ(x) = π− 1 2 x 1 2 Kµ 2
For k = 0, the expansion above thus reduces to usual Fourier expansion with
2y 1 2 KiR(2π|n + α|y) as in [39, p. 26, prop. 4.12].
As in the case of trivial multiplier, we define functions f j related to f at
each cusp, p j, of Γ by f j(z) = f |[σ j,k](z) = jσ j (z;k)−1 f (σ jz). Here σ j is the
63

cusp normalizing map of the cusp p j (cf. Chapter 1, Definition 1.1.4), and
using σ jSσ −1
j = S j together with (2.2) it easy to see that
f j(z + 1) = v(S j) f j(z) = e(α j) f j(z),
with α j ∈ [0,1) (cf. [51, p. 41]). Thus the Fourier series of f at the cusp j can
be written as
c j(n)
f j(z) =
|n + α j| Wsgn(n+α j) k 2 ,iR(4π|n + α j|y)e((n + α j)x). (2.10)
∞
∑
n=−∞
As in the case of weight 0 and Dirichlet character, we say that the cusp number
j is open or singular if α j = 0 and closed if α j = 0. If all cusps of Γ are singular
for the multiplier system v we say that v is a singular multiplier system for Γ.
Remark 2.3.1. Observe that, for the eta-multiplier and weight k, we have
α = α1 = k
12 .
2.3.1 Decomposition of the discrete spectrum
It is known (see for example [18] or [39]) that closed cusps (i.e. v(S j) = 1)
do not contribute to the continuous spectrum, and if all cusps are closed there
is only the discrete part of the spectrum left and this is spanned by the Maass
waveforms. We also know (see [39, p. 385]) that on the modular group with
weight k the least eigenvalue is
λmin = |k|
2
1 − |k|
,
2
or larger. In the case of PSL(2,Z) and k ≥ 0, F(z) = y k 2 η(z) 2k has eigenvalue
equal to λmin.
In this chapter, any eigenvalues λ ∈ λmin, 1
4 will be regarded as exceptional.
The non-exceptional eigenvalues thus satisfy 1
4 < λ0 ≤ λ1 ≤ ··· ≤
λn → ∞. One can obtain lower bounds for the eigenvalue λ0 (see [18, p. 183]),
but in light of the numerical experiments in Section 2.9.1 they are not very
effective (cf. Figure 2.2).
64

2.4 Operators
2.4.1 Conjugation and reflection
Let J and K denote the conjugation, Kz = z, and reflection, Jz = −z. Then J
and K act as involutions on the space of Maass waveforms via the operations
K f = f |K(z) = f (z).
J f = f |J(z) = f (−z).
The action of GL(2,R) on H is defined as follows; for g =
a b
c d and z ∈ H
we define
⎧
az+b
⎪⎨ cz+d , if ad − bc > 0,
g(z) =
⎪⎩ az+b
cz+d , if ad − bc < 0,
then as usual we can use the following matrix in GL(2,R) to represent the
operator J :
For T =
a b
c d we define
J =
T ∗ = JT J −1 =
1 0
0 −1
.
a −b
−c d
and then T ∗∗ = T, and T (z) |K = −T ∗ (z), meaning that also −T z = T ∗ (−¯z).
Remark 2.4.1. It is easy to verify that if f ∈ M(Γ0(N),v,k,λ), then the conjugate
K f ∈ M(Γ0(N),v,−k,λ) and J f ∈ M(Γ0(N),v ∗ ,−k,λ), where v∗ is the
multiplier system determined by
v ∗ (T ) = v(T ∗
) ·
1,
e
c = 0,
πik(1−sgn(d))
,
, forT =
c = 0,
a
c
b
d
.
Here we used the fact that for c = 0
jT ∗(Jz;k) = eikArg(−c(−¯z)+d) = e ikArg(c¯z+d) = e −ikArg(cz+d) = jT (z;k) −1 ,
while for c = 0 we have
jT ∗ (Jz;k) = eikArg(d) =
1 = jT (z;k) −1 , d > 0,
e ikπ = e 2πik jT (z;k) −1 , d < 0.
,
65

What we want to study here is the involution obtained by combining J and
K, i.e.
KJ f (z) = f |JK(z) = f (−¯z).
It is easily seen that if f has Fourier coefficients c j(n), then (by [29, p. 265(8)])
f |JK has Fourier coefficients c j(n), and we thus would like to have f and f |JK
belonging to the same space (i.e. transform according to the same multiplier
system), since then we can assume that the Fourier coefficients are real.
It is clear that if f ∈ M(Γ0(N),v,k,λ), then KJ f ∈ M(Γ0(N),v ∗ ,k,λ) so we
are left to see whether v ∗ = v or not. We consider the two multiplier systems
separately.
Eta multiplier
Suppose that v = vη, T =
a b
c d , and without loss of generality we assume
c ≥ 0. We consider the case c > 0 first:
1
2πi lnv(T ∗ ) = 1
2πi lnv
a
−c
−b
d
= k 1
+
2 2πi lnv
=
−a b
c −d
k
=
=
−a − d − 3c
+ k
− s(−d,c)
2 12c
−a − d + 3c
k
− s(−d,c)
12c
a + d − 3c
−k
− s(d,c)
12c
= − 1
lnv(T ),
2πi
so v(T ∗ ) = v(T ), and hence v ∗ (T ) = v J (T ) = v(T ). Now, if c = 0, and d > 0
we have
66
v(T ∗
) =
=
1 −b
v
0 1
−kb
e
24
= v(T ),

and if c = 0, d < 0, then
v(T ∗
) =
=
=
−1 −b
v
0 −1
k 1
e v
2 0
k kb
e + ,
2 24
b
1
and
hence v(T ∗ ) = e(−k)v(T ), and thus
Theta multiplier
v(T ) =
=
=
−1 b
v
0 −1
k 1
e v
2 0
k kb
e − ,
2 24
−b
1
v ∗ (T ) = v(T ∗ )e(k) = v(T ).
In the case of the θ-multiplier we have, for c > 0 or c = 0 and d > 0 :
a −b
vθ,k (T ∗ ) = vθ
= εd 2k
= εd 2k
=
−c d
2k −c
d
2k −1
c
2k d d
2k , ifd ≡ 1 mod 4,
i2k
c 2k
d , if d ≡ 3 mod 4,
c
d
= vθ (T ),
67

and in the case c = 0, d < 0 we have
−1 b
and
vθ (T ∗ ) = vθ
0 −1
= ε−1 2k
2k 0
−1
= (−i) 2k
= vθ (T ),
v ∗ (T ) = vθ (T ∗ )e 2kπi = i 2k (−1) 2k = (−i) 2k = vθ (T ),
hence in all cases, v ∗ θ (T ) = vθ (T ), and we conclude this section with the following
proposition.
Proposition 2.4.1. If v is either the η- or the θ-multiplier system (in the latter
case 4|N) then a basis { f1,..., fm} of M(Γ0(N),v,k,λ) can be chosen so that
each f j can be expanded in a Fourier series at ∞ with real coefficients.
Proof. We have seen that for both the theta and the eta multiplier systems the
product KJ is a conjugate-linear involution of the space M(Γ0(N),v,k,λ), and
hence we can assume that any f ∈ M(Γ0(N),v,k,λ) is an eigenfunction of
KJ with eigenvalue ε, where |ε| = 1. Note that if f (z) has a Fourier series
expansion as above, then
f |J|K(z) = f (−¯z)
= ∑
n+α=0
c(n)
W k
|n + α| 2 sgn(n+α),ν (4π|n + α|)e(x(n + α))
hence c(n) = εc(n). Finally we observe that if ε = e iθ we can look at the function
g = e i θ 2 f which then satisfies KJg = e −i θ 2 KJ f = e −i θ 2 e iθ f = g, hence we
can (after proper normalization) assume that the eigenvalue of KJ is ε = 1, and
that the Fourier coefficients are real.
Remark 2.4.2. For the sake of completeness it should be remarked that in general
one can not simultaneously take Fourier coefficients at cusps other than ∞
to be real (cf. Subsection 2.4.3 and the map ωN, which is a cusp normalizing
map for the cusp at 0 and which has eigenvalues ±i −2k ).
2.4.2 The involution ωN
As usual we define ωNz = −1
Nz , or equivalently ωN =
−1
0 √N
√ . We know that
N 0
ωN is an involution of Γ0(N), i.e. Γ0(N) = ωNΓ0(N)ω −1
n , but the question is
68

how it relates to the weight and multiplier system.
If f ∈ M(Γ0(N),v,k,λ) it is easy to see that f |[k,ωN] ∈ M(Γ0(N),v ωN ,k,λ),
and that vωN
(T ) = v ωNT ω −1
N . We also have
f |[k,ωN]|[k,ωN](z) = jωN (z;k)−1 f (ωNz) |[k,ωN]
= jωN (z;k)−1 jωN (ωNz;k) −1 f ω 2 Nz
= e −ikArg(√ Nz) e −ikArg(−1/ √ Nz) f (z)
= e −iπk f (z),
and hence if we define the operator τN by
we have that
τN f (z) = e ik π 2 f |[k,ωN](z),
τ 2 N = Id.
To prove that τN is a linear involution, what is left to verify is that v ωN = v,
which we can do in the following two cases:
a) If N = 4, and v = vθ , using θ
−1
1
2z = (−iz) 2 θ
z
2 one can prove
that vω4(A) = v(A).
b) If N = 1, and v = vη, by evaluating v ω1 on the generators of Γ0(1),
S = ω1, and T, it is immediate that v ω1
η = vη.
And we conclude:
Proposition 2.4.2. If N = 1 and v = vη, or N = 4 and v = vθ the operator
defined by
τN : M(Γ0(N),v,k,λ) → M(Γ0(N),v,k,λ),
τN f (z) = e ik π 2 f |[k,ωN] = e −ik(Argz− π 2 ) f
is a linear involution, i.e. τ2 N f = f .
−1
,
Nz
This means that the operator τN is a linear involution of M(Γ0(N),v,k,λ),
and hence has eigenvalues ±1. Note that in the case N = 4, if τN f (z) = ± f (z),
then
f2 = f |ωN = e−ik π 2 τN f = ±e −ik π 2 f ,
which in turn means that the Fourier coefficients at the cusp at 0 satisfy
c2(n) = ±e −i π 2 k c1(n) = ±(i) −k c1(n). (2.11)
69

2.4.3 The operator σ4
For N = 4 and θ-multiplier system we define the operator σ4, by
z + j
.
σ4 f (z) = ∑
j mod 4
f
This can be viewed as a “partial” Hecke operator on Γ0(4). (cf. [53, p.
195, (0.8)] and [56, p. 250]). We have to verify that σ4 is an operator on
M(Γ0(4),vθ ,k,λ). Suppose that A =
a b
4c d ∈ Γ0(4), and with the notation
Vj =
1 j
, we get
0 4
(σ4 f ) |[A,k](z) = jA(z;k) −1 ∑
j mod N
f (VjAz)
Here we used
jB(Vlz;k) = exp
= jA(z;k) −1 ∑
l mod N
f
4
∗ ∗
16c d − 4lc
Vlz
{l is uniquely defined mod 4 by al ≡ b + jd mod 4}
= jA(z;k) −1 ∑ ε
l mod N
2k
2k 16c
d
vθ (B) jB(Vlz;k) f (Vlz)
d − 4lc
= ε2k
16c
d ∑
f (Vlz).
d − 4lc
= jA(z;k).
l mod N
ikArg 16c
z + l
4
+ d − 4lc = exp[ikArg(4cz + d)]
Hence, what is left is to check that
16c c c
d−4lc = d−4lc = d .
The sign of Nc is the same as the sign of N2c and we might assume without
loss of generality that c > 0. By virtue of (2.7) we may also assume that c is
odd since we can always pull out a factor of 2 at the beginning. For the reciprocity
relations applicable to our quadratic residue symbol, see Proposition
2.2.2:
70
c
d − 4lc
=
=
=
d − 4lc
c−1
(−1) ( 2 )(
c
d−4lc−1
2 )
d
c−1
(−1) ( 2 )(
c
d−1
2 )−l(c−1)c d
c−1
= (−1) ( 2 )(
c
d−1
2 )
c
,
d

(note that c(c − 1) is even). We have shown that
(σ4 f ) |[A,k](z) = ε 2k
d
c
2k d
∑
l mod N
and thus σ4 : M(Γ0(4),vθ ,k,λ) → M(Γ0(4),vθ ,k,λ).
f (Vlz) = vθ,k(A)σ4 f (z),
Remark 2.4.3. Following Kohnen [56] or Katok-Sarnak [53] we define the
operator L acting on M(Γ0(4),vθ , 1
2 ,λ) by
L = τ4σ4.
It can be shown that this operator is self-adjoint, commutes with ∆ 1 and have
2
the eigenvalues 1 and 1
2 .
2.4.4 Maass operators
So far, the operators we have seen act on spaces of Maass waveforms of a
given weight and multiplier system.
We will show that we may limit the range of weights k to investigate to k ∈
[0,6]. For this we will use the Maass lowering and raising operators, E ± k ,which
raise or lower the weight of a Maass waveform by units of 2. They are defined
by
E ± k
∂ ∂ k
= iy ± y +
∂x ∂y 2 ,
and using the relation between the Whittaker function and the confluent hypergeometric
function together with the transformation formulas [29, p. 258,
(10)] (see also: [63, p. 183 (middle)] and [64, p. 302 lines -3 and -1]) we see
that for k > 0 (use the abbreviation 4π|n + α|y = Y )
(Y )e((n + α)x) = −Wk+2
E + k
E − k
E + k
E − k
W k 2 ,iR
W k − 2 ,iR (Y )e((n + α)x)
=
2
k(k + 2)
4
,iR (Y )e((n + α)x) if n + α > 0,
+ 1
+ R2
4
W k+2 − 2 ,iR (Y )e((n + α)x)
if n + α < 0,
W k − 2 ,iR (Y )e((n + α)x) =W k−2 − 2 ,iR (Y )e((n + α)x)
k(k − 2)
W k
2 ,iR (Y )e((n + α)x) = − +
4
if n + α < 0, and
1
+ R2
4
ifn + α > 0.
Wk−2
2 ,iR (Y )e((n + α)x)
To verify that they respect the weight k-action we rewrite them in the follow-
71

ing form
E + k
E − k
and also use the formulation:
∂ k
= (z − z) +
∂z 2 ,
∂ k
= −(z − z) +
∂z 2 ,
e −iArg(cz+d) =
1
cz + d 2
.
cz + d
It is then easy to see (cf. [63, p. 178]) that
E +
k
f|[k,A](z)
= (z − z) ∂
k
+
∂z 2
k
cz + d 2
f (Az)
cz + d
k+2
−k cz + d 2 c
=(z − z)
f (Az)
2 cz + d (cz + d)
k
cz + d 2 1
+
cz + d (cz + d) 2 f ′
(Az) + k
k
cz + d 2
f (Az)
2 cz + d
k
cz + d 2 k c(z − z)
(z − z)
=
1 − f (Az) +
cz + d 2 (cz + d)
k+2
cz + d 2
=
cz + d
(z − z)
|cz + d| 2 f ′ (Az) + k
f (Az)
2
k+2
cz + d 2
=
cz + d
′ k
Az − Az f (Az) + f (Az)
2
= E +
k f |[k+2,A] .
(cz + d) 2 f ′ (Az)
It is known that E ± k maps M(Γ0(N),v,k,λ) into M(Γ0(N),v,k±2,λ), and that
the composition
E ∓ k±2 E± k : M(Γ0(N),v,k,λ) ↦→ M(Γ0(N),v,k,λ)
is just multiplication by a constant, which is readily seen to be nonzero anytime
λ > 1
4 . Hence E± k acts bijectively on the spaces corresponding to nonexceptional
eigenvalues, i.e. they are always bijections for λ > 1
4 .
2.4.5 Maass operators and the symmetry about k = 6
First of all, observe that E ± k
only change the weight and not the multiplier
system v, but in view of the remark after Def. 2.2.2 it is clear that v = v 2k
η is
72

also a multiplier system of weight k + r for any r ∈ 2Z, and with the notation
it is clear that
v (r)
η,k
and
= v2(k+r)
η
E + k
E − k
(r)
(r−2)
: M(Γ0(N),v η,k ,k,λ) → M(Γ0(N),v η,k+2 ,k + 2,λ),
(r)
(r+2)
: M(Γ0(N),v η,k ,k,λ) → M(Γ0(N),v η,k−2 ,k − 2,λ).
Our main purpose is to investigate the locations of Maass waveforms on
the modular group when the weight and multiplier system are varied. That
is, we would like to investigate the space M(Γ0(1),v (r)
η,k ,k,λ) for all k ∈ R
and r ∈ {0,2,4,6,8,10}. To reduce the problems we use the fact that v (r)
η,k are
all 24th-roots of unity. Suppose that λ > 1
4 so the the lowering and raising
operators act bijectively, then using the notation (k,v (r)
η,k ) to denote that space
M(Γ0(1),v (r)
η,k ,k,λ) we have:
• A composition of Maass operators which raises or lowers the weight by 12
will preserve the multiplier system. Hence it is no restriction to assume
that k ∈ [0,12].
• Conjugation maps f ∈ (k,v (r)
η,k ) to K f ∈ (−k,v(−r) η,−k ) = (−k,v(12−r)
η,−k ), which
can in turn be mapped to a function g ∈ (12 − k,v (12−r)
η,12−k ) by raising operators,
so there is no restriction to assume k ∈ [0,6] and r ∈ {0,2,4,6,8,10}.
• But by using the raising operator repeatedly we can map v (0)
η,k to any v(r)
η,k
and hence it is no restriction to also assume r = 0.
We are thus justified in our choice of restricting the investigation to the spaces
M(Γ0(1),v,k,λ) for k ∈ [0,6] and v = v (0)
η,k
2.4.6 Hecke operators
= v2k
η .
We know that Hecke operators play an important role in the understanding of
the theory of both modular forms and Maass waveforms at integer weights.
For general real weights, the Hecke operators are not important, but we will
begin with the general definition anyway, and then we will consider the case
of weight 1
2
and the theta multiplier system, where the action of the Hecke
operators T p 2 are well understood.
This discussion is based on [93], and see also [103].
Let Γ ⊂ PSL(2,R) be cofinite and suppose v : Γ→ S 1 is a multiplier system
of weight k ∈ R. The commensurator of Γ, comm(Γ), in PSL(2,R) is defined
as
comm(Γ) = α ∈ PSL(2,R)|αΓα −1 ∩ Γ has finite index in Γ and αΓα −1 .
73

We know that the Hecke operators are associated with the members of the
commensurator, or actually with the double cosets, ΓαΓ, for α ∈ comm(Γ).
Fix α ∈ comm(Γ). It can be shown that if the multiplier system v satisfies
v(g) = v α (g), ∀g ∈ Γ ∩ α −1 Γα, (2.12)
then we can define an associated multiplier system, vα, on the double coset:
by setting
vα : ΓαΓ → S 1 ,
vα (g1αg2) = σk (g1α,g2)σk (g1,α)v(g1)v(g2),
for all g1,g2 ∈ Γ. It might be the case that there exists an associated multiplier
system of the multiplier system W = χv, where χ is a character on ΓαΓ, even
though there does not exist an associated multiplier system of v itself. Suppose
that vα exists as above, and that ΓαΓ = ∪ d i=1 Γαi. We then define the Hecke
operator T v α,k : M Γ,v,k,λ → M Γ,v,k,λ by
T v α,k f (z) =
=
d
∑ vα (αi) f |[αi,k](z)
i=1
d
∑ vα (αi) jαi
i=1
(z;k)−1 f (αiz).
When Γ is a congruence subgroup, one usually constructs Hecke operators
corresponding to positive integers n, in which case α =
1 0
0 n and αi =
a b
0 d
with ad = n, d > 0, b mod d. The Hecke operator T v
n,k is then defined by
T v
n,k f (z) = n k 2 −1
d−1
a b az + b
∑ ∑ vα
f .
0 d d
ad=n b=0
The Theta multiplier
Consider the case Γ = Γ0(4), k = 1
2 and v = vθ . Let n be a positive integer and
α =
1 0
0 n . Then g ∈ Γ0(4) ∩ α−1
a nb
Γ0 (4)α can be written g = c/n d , and
αgα−1 =
a b , with ad − bc = 1 and c ≡ 0 mod 4n. It is easy to verify that
c d
vα (g) = v(αgα−1 ), and
c
n
v(g) = εd
d
= εd
c
d
n
−1 = v(αgα
d
−1 )
n
−1 ,
d
and hence vα (g) = v(g) if and only if
n
d = 1. Since (2.12) implies this condition
for all odd integers d it is clear that the extension vα exists if and only
if n is a perfect square.
74

Suppose now for simplicity that n = p2 , with p a prime number = 2. It
is easy to verify that in this case the associated multiplier system is given
by vα (g1αg2) = vα (g1)vα (g2), and the different coset representatives are
1 b
given by 0 p2
,b = 0,..., p2
− 1, p2 0
p b
, and
0 1
0 p , b = 0,..., p − 1. By
factorization of these representatives we see that
p
vα
2
b
1 0
= vα
0 1
0 p2
1 b
= 1,
0 1
p2
0
p2
−t 1 0 p2
d t
vα
vα
and hence
0 1
= vα
4 d 0 p2 −4 1
= p 2 d + 4t = 1
p
= v
2
−t
4 d
= εd = 1,
p b
0 p
= vα
T vθ
p2 , 1 f (z)
2
= p − 3 =
d−1
2 ∑ ∑ vα
ad=p2 b=0
p − 3
p2−1 2 ∑ vα
b=0
vα
= p − 3 2
= p − 3 2
1 0
4xp 1
1 0
0 p 2
p b
−4x D
= {pD + 4bx = 1}
p b
−4x
= v
= εD
−4x D
D
−4x −4x −4x
= εp
= εp
pD p
p
4xb −b −b b
= εp
= εp = εp ,
p p p p
a b az + b
f
0 d d
1 b
0 p2
z + b
f
p2
+
p2
0
f
0 1
p 2 z
p−1
p b
+ ∑ vα
f z +
b=0 0 p
b
p
p2−1
z + b
∑ f
b=0 p2
+ f p 2 z p−1
b
+ ∑ εp f z +
b=0 p
b
p
z + b
f
p2
+ f p 2 z p−1
b
+ εp ∑ f z +
p
b
p
.
p 2 −1
∑
b=0
b=0
75

This agrees with the corresponding results in [90, pp. 450–451]. To see that
the operator obtained is the same as in [53] (up to a constant multiple) we look
at the action on the Fourier coefficients a(n) of f ∈ M Γ0(4),vθ , 1
2 ,λ , i.e.
with the abbreviation W(x) = W1
4 sgn(x),iR (4π|x|), and
we have
and note that
and
T vθ
p2 , 1 f (z) = p
2
− 3 2
p−1
∑
b=0
b
e
p
f (z) = ∑ n=0
+ ∑ n=0
+ εp
= p − 3 2
+ ∑ n=0
+ ∑ n=0
p 2 −1
∑
b=0
a(n)
W(ny)e(nx),
|n|
∑
n=0
a(n) ny
W
|n| p2
n(x + b)
e
p2
a(n)
W(np
|n| 2 y)e(np 2 x)
p−1
∑
b=0
p2−1 ∑ e
b=0
nb
p
(cf. [61, IV, §3]) which gives us 1
b a(n)
p ∑ W(ny)e nx +
n=0 |n| nb
p
a(n)
∑ W
n=0 |n|
ny
p2
e
a(n)
W(np
|n| 2 y)e(np 2 x)
a(n)
W(ny)e(nx)εp
|n|
nx
p2 p2−1 p−1
∑
b=0
nb
p2
p
=
2 , p2 |n,
0, p2 ∤ n,
∑ e
b=0
b
e
p
nb
p2
nb
p
,
p−1
n nb nb
=
p ∑ e
b=0 p p
= √
n
p
p
1, p ≡ 1 mod 4,
i, p ≡ −1 mod 4,
= √
n
p εp,
p
1 For p = 2, we use a definition analogous to [90, p. 450] and derive an expansion analogous to
[90, thm. 1.7].
76

T vθ
p2 , 1 f (z) =
2
1
√
p ∑
n=0
1
|n|
a(p 2
n
n) + a
p2
+ p − 1 2
n
a(n) W(ny)e(nx).
p
It is easy to see that our Hecke operator T vθ
p2 , 1 =
2
1 √
pTp2, where Tp2 is the corresponding
Hecke operator defined in [53, p. 199]. Observe that our Fourier
coefficients a(n) = √ n×Katok-Sarnak’s Fourier coefficients b(n).
As usual, we can look at Hecke eigenforms in M Γ0(4),vθ , 1
2 ,λ , and one
can prove (cf. [90, p. 453]) that if f = ∑ a(n)
√ W(ny)e(nx) is an eigenfunction
n
of T vθ
p2 , 1 for all primes p and if t is square free, then
2
a(tm 2 )a(tn 2 ) = a(t)a(tm 2 n 2 ), for(m,n) = 1.
2.4.7 Lifts at weight 1 and Fourier coefficients
As a general rule, in cases with nontrivial multiplier system, it is hard to see
precisely which Hecke operators (if any) are present and what the corresponding
multiplicative relations are. In this section we discuss a simple reduction
trick which allows one to get some multiplicative relations in a case where
k = 1 and α = 0 (cf. (2.9)).
To this end, consider any Maass waveforms f ∈ M(Γ0(1),v2 η,1,λ). This
implies that, for ad − 144bc = 1, we have (see Prop. 2.2.3)
f (Az) = jA(z;1)v 2
−1
η (A) f (z) = jA(z;1) f (z),
d
where A =
a 12b
12c d , i.e. f ∈ M(Γ12,
−1
d ,1,λ). We now follow a standard
idea to “lift” f to a function of weight 1 on Γ0(144) with a Dirichlet character.
Indeed, set
g(z) = f (12z).
Then, with
A =
a b
144c d
∈ Γ0(144),
77

we see that
az + b 12az + 12b
g(Az) = g
= f
144cz + d 144cz + d
a(12z) + 12b
= f
12c(12z) + d
= e iArg(144cz+d)
−1
f (12z)
= e iArg(144cz+d)
d
−1
d
g(z),
i.e. g ∈ M(Γ0(144),
−1
d ,1,λ). The advantage of this transformation is that
the theory of Hecke operators is well understood in the case of a Hecke congruence
subgroup, Γ0(N), together with a Dirichlet character χ. In fact, the
Hecke operators on M(Γ0(144), χ,1,λ) are the same as at weight 0:
az + b
Tn f = ∑ ∑ χ(d) f ,
d
ad=n
d>0
b mod d
and the multiplicativity relation is the usual (cf. [69, p. 37 (para 3) and §4.5.]):
TnTm = ∑
d|(m,n)
χ(d)dTmn .
d2 We will now consider how the existence of these Hecke operators gives rise
to relations between the Fourier coefficients of f . Assume that we have
∞ a(n)
f (z) = ∑ n + 1
4π
1
n +
12
y
e n + 1
x ,
12
then
f (12z) =
n=−∞
∞
∑
n=−∞
= √ 12
a(n)
n
1 +
∞
∑
n=−∞
12
W1 2 sgn(n),iR
12
W1 2 sgn(n),iR
and hence the Fourier expansion of 1
√12 g is
78
∑
n≡1 mod 12
4π
1
n +
12
12y
e n + 1
12x
12
a(n)
W1
|12n + 1| 2 sgn(n),iR (4π |12n + 1|y)e((12n + 1)x),
c(n)
W1
|n| 2 sgn(n),iR (4π|n|y)e(nx),

where
n − 1
c(n) = a .
12
The multiplicativity relations for the Hecke operators imply similar relations
for the normalized coefficients c(n)/c(1) :
−1
mn
c
d d2
, (2.13)
c(m)c(n) = c(1) ∑
0

and that if g ∈ M(Γ0(144),
−1
d ,1,λ) satisfy Λg = εg with ε = ±1, and have
Fourier coefficients c(n) as above, then
c(−n) = ±iRc(n).
Using this in combination with (2.13) we get the following relation
c(−n)c(−m) = −R 2 c(1) ∑
00
−1
d
c
mn
d 2
valid for −n ≡ −m ≡ 1 mod 12 (which implies mn ≡ 1 mod 12). If we change
to a’s we get that for any positive integers m,n we have
−1 (12m − 1)(12n − 1) − d2 a
d
12d2
.
And to get a simpler relation (that is easy to verify numerically) we set
m = −1, and we get two cases. If (12n − 1,11) = 1 then
and if (12n − 1,11) = 11 then
,
a(−n) = −R2a(0) a(11n − 1), (2.17)
a(−1)
a(−n) = −R2 a(0)
a(−1)
a(11n − 1) − a
n − 1
11
. (2.18)
We have not seen relations of type (2.16)-(2.18) earlier in the literature. For
some numerical examples, see Tables 2.2 and 2.3.
Remark 2.4.4. It is clear that not all forms on M(Γ0(144),
−1
d ,1,λ) correspond
to forms on M(Γ0(1),v2 η,1,λ) (look at the Weyl’s law for the two cases).
From experiments we see that M(Γ0(1),v2 η,1,λ) seems to contain cosine forms
of CM-type as well as other “generic” forms. See Table 2.4.
2.5 Oldforms
Clearly the usual f ∈ M(Γ0(M),v,k,λ) ⇒ f ∈ M(Γ0(N),v,k,λ) holds whenever
M|N. The obvious notation is to call such a function f an oldform on
M(Γ0(N),v,k,λ). We will look at one example of an oldform.
80

2.5.1 Γ0(N) with N prime and η-multiplier
The cusps are p1 = ∞ and p2 = 0, with cusp normalizing maps and stabilizers
(i.e. generators of the stabilizer subgroups) of σ1 = Id, S1 = S and
σ2 =
0
N
−1
0
S2 = σ2Sσ −1
2 =
1
−N
0
1
.
The shifts are α1 = k
and let T = 0 −1
1 0
12 = Nα1.
Suppose that f ∈ M(Γ0(1),v,k,λ),
and dN =
N 0
0 1 so σ2 = T dN then
12 and α2 = Nk
f2(z) = f |σ2 (z) = jσ2 (z;k)−1 f (σ2z)
= jσ2 (z;k)−1 f (T dNz)
= jσ2 (z;k)−1 jT (dNz;k)v(T ) f (dNz)
= e −ikArgz e ikArgz v(T ) f (dNz)
= e −ikπ v(T ) f (Nz)
= e −i π 2 k f (Nz) = (−i) k f (Nz).
In terms of the Fourier series this means that
f2(z) =
c2(n)
|n + α2| Wsgn(n+α2) k 2 ,µ(4π|n + α2|y)e((n + α2)x)
f (Nz) =
=
=
=
∞
∑
−∞
∞
∑
−∞
∞
∑
−∞
∞
∑
−∞
∞
∑
m=−∞
c1(n)
|n + α1| Wsgn(n+α1) k 2 ,µ(4π|n + α1|Ny)e((n + α1)Nx)
c1(n)
|n + α2
N |
W α
sgn(n+ 2
N ) k 2 ,µ(4π|Nn + α2|y)e((Nn + α2)x)
√
Nc1(n)
|Nn + α2| Wsgn(Nn+α2) k 2 ,µ(4π|Nn + α2|y)e((Nn + α2)x)
√
m Nc1 N
|m + α2| Wsgn(m+α2) k 2 ,µ(4π|m + α2|y)e((m + α2)x),
where as usual c(r) = 0 if r ∈ Z, and since f2(z) = (−i) k f (Nz) we must have
the following relation:
c2(Nn) = (−i) k√ Nc1(n).
81

2.6 The Eisenstein series
In case there is a cusp p j at which the multiplier system is singular (i.e.
v(S j) = 1) we have a continuous spectrum: [ 1
4 ,∞) (with multiplicity equal
to the number of singular cusps), and in general we can not say much about
the embedded discrete spectrum in [ 1
4 ,∞).
Singular cusps can appear either because of the multiplier system, e.g. the
θ-multiplier system, or because the “twist” α j is an integer (remember v(S j) =
e 2πiα j ), e.g. on the modular group with the η-multiplier system and weight
k = 12l.
Remember that Maass waveforms are part of the discrete spectrum, but as
we continuously “turn off” the multiplier system, i.e. for v = v 2k
η we let k → 0,
the continuous spectrum will emerge in the limit. For this reason we want to
review some details concerning the Eisenstein series on the modular group,
i.e. for the rest of this section Γ = PSL(2,Z).
2.6.1 Weight 0
At weight 0 and singular character χ, the continuous spectrum of ∆ = ∆0 is the
interval [ 1
4 ,∞) and the eigenpacket is given by the Eisenstein series E(z;s; χ)
defined by
E(z;s; χ) = ∑
T ∈Γ∞\Γ
χ(T −1 )(ℑ(T z)) s ,
where Γ∞ = [S]. (See [39, p. 65]). From proposition 6 of the same page we get
E(z;s; χ) = y s + ϕ(s)y 1−s ∑ ϕm(s)
m=0
√ yK 1 s− (2π|m|y)e(mx)
2
here if W∞ =
a b
c d are the double coset representatives, i.e. Γ = ∑Γ∞W∞Γ∞
then
ϕ(s) = √ π Γs − 1
2
∞ )
Γ(s)
1
,
|c| 2s
and
82
ϕm(s) = 2πs |m| s− 1 2
Γ(s)
∑ χ(W
W∞∈Γ∞
−1
∑ χ(W
W∞∈Γ∞
−1
∞ ) 1
|c|
2s e
md
.
c

For trivial character χ = 1 we get (see page 76 of the above)
ϕ(s) = √ π Γs − 1
2 ζ (2s − 1)
Γ(s) ζ (2s)
ϕm(s) = 2πs |m| s− 1 2
Γ(s)
σ1−2s(|m|)
.
ζ (2s)
Hence we can see that for s = 1
2 + iR , the n−th Fourier coefficient of E(z;s)
is given by
1
c(n) = ϕ + iR = K · |n|
2 iR σ−2iR(|n|), (2.19)
= K · |n| iR ∑
d||n|,d>0
d −2iR
where K = K(R) is a constant dependent on R. First we observe the simple
formula for prime coefficients (p is positive prime)
c(p) = K · p iR (1 + p −2iR )
= 2K · cos(Rln p)
so we can compute quotients of c(p) and compare this with corresponding
quotients for our forms of weight = 0.
2.7 Half integer weight
We know that the θ−function is an automorphic form (not a cusp form) of
weight 1
2 on Γ0(4), hence we consider Γ0(4) together with the θ−multiplier
system (cf. section 2.2.3).
2.7.1 The Shimura correspondence
Introduction – The Holomorphic Case
We will consider the Shimura correspondence only in the particular case of
trivial character and level a square-free multiple of 4. Let Sk(M) denote the
space of holomorphic cusp forms of weight k ∈ Z (and trivial multiplier) on
Γ0(M), and let S 1 k+ (4N) denote the space of holomorphic cusp forms of
2
weight k + 1
2 , k ∈ Z and multiplier vθ , on Γ0(4N). The Shimura correspondence
is a correspondence between the space S 1 k+ (4N) and spaces S2k(N
2
′ )
for certain integers N ′ |4N (e.g. N ′ = 2N or N).
The map from S 1 k+ to S2k was first constructed by Shimura [90] and later
2
was constructed by Shintani [92]. The for-
an adjoint map from S2k to S k+ 1 2
83

mer uses a Dirichlet-series and the latter uses an integral against a thetafunction.
Both these maps commute with the Hecke operators that are acting
on S2k(N) and S 1 k+ (4N) respectively. Kohnen [56, 57] proved that for
2
N odd and square-free, the correspondence is a bijection between the newforms
on S2k(N) and a certain subspace, V + ⊆ S 1 k+ (4N). The subspace V
2
+
is composed of Hecke eigenfunctions whose Fourier coefficients, c(n), satisfy
certain vanishing properties; namely, c(n) = 0 for n ≡ 2,3 mod 4. (Here we
shall mainly be interested in N = 1.)
Extension to the Non-holomorphic case
The extension of the Shimura correspondence and Kohnen’s result to spaces
of Maass waveforms has been investigated by e.g. Sarnak [87], Hejhal [37],
Duke [28], Katok-Sarnak [53], Kojima [58, 59], Biró [13] and Arakawa [4].
It is not clear exactly what the range and domain of the correspondence are
in the Maass case.
Define the two spaces U + ⊆ M(Γ0(1),0,2R) and V + ⊆ M Γ0(4),vθ , 1
2 ,R
of Hecke normalized Maass waveforms. Here U + consists of the even (cosine)
Hecke eigenforms on Γ0(1) and V + consists of the Hecke eigenforms (of the
) which are invariant under the operator L = τ4σ4 (cf.
Hecke operators T vθ
p 2 , 1 2
Remark 2.4.3). As in the holomorphic case one can show that the condition
L f = f is equivalent to the statement that the Fourier coefficients of f satisfy
c(n) = 0 for n ≡ 2,3 mod 4 (cf. [56, prop. 2].
From [53, prop. 4.1 and 2.3] we know that to each φ ∈V + there corresponds
a lift f = F(φ), which is either ≡ 0 or in U + . (The “0” occurs if and only if
the first Fourier coefficient of φ is zero.) Conversely it is also true that for
each f ∈ U + there exists φ ∈ V + such that f = F(φ).
By combining what is known from the holomorphic and non-holomorphic
cases, we would also expect to find a correspondence between even newforms
on Γ0(2) of weight 0 and spectral parameter 2R and certain (other, non V + )
half integer forms on Γ0(4) with spectral parameter R, and indeed we found
such a correspondence in our numerical investigations which we present in
Subsection 2.9.4.
Fourier coefficients
We will now see how to relate the Fourier coefficients of the half integral and
corresponding integral weight form.
There are formulas in [53] and [58] connecting the Fourier coefficients of
half integral and integral weight, but due to differences in normalizations it
might be clearer to go back to Shimura ([90]) and start there.
Suppose first that we have two Hecke eigenforms: f and φ, where f (z) ∈
M(Γ0(2),0,2R) + (the space of even newforms) and φ(z) ∈ M Γ0(4),vθ , 1
2 ,R .
84

Further suppose that their respective Fourier series expansions with respect to
the cusp at ∞ are
and
f (z) =
φ(z) =
∞
∑
n=−∞
n=0
∞
∑
n=−∞
n=0
A(n) √ yK2iR (2π|n|y)e(nx),
a(n)
W1
|n| 4 sgn(n),iR (4π|n|y)e(nx),
where we use the normalization a(1) = 1 and A(1) = 1.
By [53, eq. (4.2)] and the analogy with what happens in the holomorphic
case (cf. [90, pp. 453, 458]), when f is the lift of φ, one has the following:
Natural Conjecture. Let f ∈ M(Γ0(2),0,2R) + be the Shimura lift of φ ∈
M Γ0(4),vθ , 1
2 ,R . Suppose that f has Fourier coefficients {A(n)} and that φ
has Fourier coefficients {a(n)}(with respect to ∞). Let t ∈ Z + be square free
such that a(t) = 0. Then, for each n ∈ Z + , we have:
A(n) = ∑
kd=n
k>0
χ ′ t (k)
√ k
where χ ′ t is the quadratic residue symbol t
·
448(prop. 1.3), 458 (main theorem), 474 (line -11)].)
Outline of proof. Identify coefficients in the relation
∞
∑ A(n)n
n=1
−s = c · L
s + 1
2 , χ′ t
a td2 , (2.20)
a(t)
considered mod 4t. (Cf. [90, pp.
∞
∑ a
n=1
tn 2 n −s , (2.21)
which can be seen as the Maass waveform equivalent of [90, p. 458] and [72,
p. 159], or alternatively as an extension (to newforms) of [53, eqs. (0.15) and
(4.2)].
To obtain this relation, it seems that it is possible to follow the construction
in [53], but using the theta function from Example 3 in [72, p. 151–153] instead
of θ(z,g). In our setting this amounts to twisting the theta function by the
character
t
· , and apart from different transformation rules all calculations in
[53] should go through.
The main difference in the argument is that instead of starting with a function
φ(z) on Γ0(4), one starts with φ(tz) on Γ0(4t). Cf. [90, p. 474]. Also see
[54, prop. 4.1 and thm. 5.1(2)]
85

In the earlier (oldform) setup of [53], (2.21) should hold with χ ′ t considered
mod t. Likewise for (2.20).
This point requires one clarification, however. Namely: since weight 0
oldforms on Γ0(2) occur in pairs, the Shintani map can be expected to give
rise to a corresponding pair of weight 1
2 forms on Γ0(4). In a suitable basis,
one of these should lie in the space V + ; the other should not but should still
for p = 2. Cf. [69, lemmas 4.6.2
have the same Hecke eigenvalues under T vθ
p 2 , 1 2
and 4.6.10]. In light of this, the second form can still be expected to satisfy
(2.20), but only for (n,4) = 1.
The preceding (partly conjectural) statements are backed by strong numerical
evidence. Part of this evidence can be found in Table 2.8. Further conjectures
(backed by numerics) concerning the Shimura correspondence can be
found in Section 2.9.4.
See [19, §4.1] and [21, p. 633], and [91, pp. 502 (bottom) – 504 (top)] for
some additional perspectives.
2.8 Some Computational Remarks
A priori there are basically only two modifications that need to be done in
order to make the weight zero algorithm work in the general case:
• the K-Bessel function needs to be replaced with the Whittaker function,
and
• the automorphy condition needs to incorporate the multiplier system.
The first modification, although trivial in theory is the most complex in terms
of the numerics involved. There was no efficient algorithm for the Whittaker
function in the literature and we had to develop a new algorithm. We used the
integral representation (cf. [39, p. 375 (top)])
2x ∞
Wk,iR(2x) = e
π
−xcosht
Ψ
0
−k; 1
;x(1 + cosht)
2
cosh[iRt]dt,
where Ψ is a confluent hypergeometric function together with a stationary
phase method to develop a robust and efficient algorithm. This algorithm is
in essence similar to Hejhal’s algorithm (cf. [42]) for the K-Bessel function,
KiR(x), and the generalization of it made by Avelin (cf. [9]) to a complex
argument, Ks(x), s ∈ C. The details of this algorithm are described in Chapter
4.
To modify the automorphy condition is trivial. Observe that f j(z) = f |σ (z) = j
jσ j (z;k)−1 f (σ jz), and with z∗ −1
j = σI( j) UI( j)Tjσ jz (cf. p. 35) the automorphy
condition (1.19) now becomes:
86

f j(z) = jσ j (z;k)−1 f (σ jz) = jσ j (z;k)−1 f (T −1 −1
j UI( j) σI( j)z ∗ j)
= jσ j (z;k)−1v(T −1 −1
j UI( j) ,k) jT −1
j U−1 (σI( j)z
I( j)
∗ j;k) f (σI( j)z ∗ j)
= jσ j (z;k)−1 j −1
Tj U−1 (σI( j)z
I( j)
∗ j;k) jσ (z I( j) ∗ j;k)v(T −1 −1
j UI( j) ,k) fI( j)(z ∗ j).
The entire setup of the Phase 1 algorithm (i.e. locating eigenvalues and
computing the first few Fourier coefficients) goes through exactly as in the
case of weight 0 (cf. Section 1.3.2) with some trivial modifications. For the
sake of completeness we will give an outline of the modified algorithm.
Consider a Maass waveform, f , of weight k and multiplier system v. At
each cusp it has a Fourier series
fi(z) =
∞
∑
−∞
n+α i =0
ci(n)
|n + αi| W sgn(n+αi) k 2 ,iR(4π|n + αi|y)e((n + αi)x),
and since the Whittaker function is ultimately exponentially decaying, given
an ε > 0, there exists a constant (depending on y and ε), M(y), such that
fi(z) = ˆfi(z) + [[ε]],
where we use [[ε]] to denote a constant of magnitude less than ε. The truncated
Fourier series
ˆfi(z) =
M(Y )
∑
−M(Y )
n+α i =0
ci(n)
|n + αi| Wsgn(n+αi) k 2 ,iR(4π|n + αi|y)e((n + αi)x),
is now viewed as a discrete Fourier transform, and if we take the inverse
transform over the horocyclic points: zm = xm + iY , 1 − Q ≤ m ≤ Q, where
− m) we get:
xm = 1 1
2Q ( 2
1
2Q
= 1
2Q
=
Q
∑
1−Q
Q
∑
1−Q
M(Y )
∑
−M(Y )
l+α i =0
ˆfi(zm)e(−(n + α j)xm) =
M(Y )
∑
−M(Y )
l+α i =0
ci(l)
|l + αi| Wsgn(l+αi) k 2 ,iR(4π|l + αi|Y )e((l + αi)xm − (n + α j)xn)
ci(l)
|l + αi| Wsgn(l+αi) k 2 ,iR (4π|l + αi|Y ) 1
2Q
= ci(n)
|n + α j| W sgn(n+α j) k 2 ,iR (4π|n + α j|Y ),
Q
∑
1−Q
e((l + αi)xm − (n + α j)xn)
87

where we used the fact that
Q
1
2Q ∑ e((l + αi)xm − (n + α j)xn)
1−Q
=
(l + αi − n − α j)
e
×
4Q
1
2Q ×
Q
× ∑ e −(l + αi − n − α j) m
2Q
[since α ∈ [0,1)] = δnl.
1−Q
= δ{l + αi = n + α j}
Now we also have fi(zm) = χmi fI(m,i)(z∗ mi ), where
χmi = jσi (zm;k) −1 j −1
Ti U−1
I(m,i) σI(m,i) (z∗mi;k)w(T −1
i
Hence
where
i j
Vnl =
= 1
2Q
= 1
2Q
=
U −1
I(m,i) ,σI(m,i))v(T −1
i
ci(n)
|n + αi| Wsgn(n+αi) k 2 ,iR(4π|n + αi|y)
κ
∑
j=1
We then define ˜V
i j
nl by
˜V
i j i j
nl = V
Q
∑
1−Q
fi(zm)e(−(n + αi)xm) + [[ε]]
Q
∑ χmi fI(m,i)(z 1−Q
∗ mi)e(−(n + αi)xm) + [[ε]]
M0
∑
l=−M0
n+α i =0
1 1
|l + α j| 2Q
c j(l)V
i j
nl
+ 2[[ε]],
U −1
Q
∑ χmiWsgn(l+α j)
1−Q
I(m,i)= j
k 2 ,iR(4π|l + α j|y ∗ m j) ×
×e (l + α j)x ∗
m j e(−(n + αi)xm).
nl − δnlδ ji
1
|n + αi| Wsgn(n+αi) k 2 ,iR(4π|n + αi|Y ).
If we now neglect the error ε we have a linear system
I(m,i) ,k).
CV = 0, (*)
which must be satisfied by the Fourier coefficients of f . Here V is the matrix
˜V
i j
nl and C is the vector ci(n), both depending on R and Y . The basic idea
88

of Phase 1 is now to locate eigenvalues R together with sets of corresponding
Fourier coefficients, ci(n), by solving (*) repeatedly for different R’s, and
seeking those values of R for which the solution vectors C = C(R,Y ) are stable
under changes of Y . That is, if C(R,Y1) ≈ C(R,Y2) (for Y1and Y2 < Y0,
for some suitable constant Y0) we take it as an indication that the R is close
to an eigenvalue and that the components of C are close to the corresponding
Fourier coefficients. For more details and justifications see Section 1.3.2.
We note that the Phase 2 algorithm of Section 1.3.3 (i.e. computation of
a larger set of Fourier coefficients) also generalizes to non zero weight in a
similar manner.
2.9 Numerical results
The experimental excursions have been directed towards three essentially different
subjects, but, in each, we have worked in an exploratory spirit.
• First we tried to get an over-all picture of the distribution of small to middlerange
sized eigenvalues on PSL(2,Z) (and the eta multiplier) for “large”
weights, e.g. R ∈ [0,14] and k ∈ [0.1,6].
• Second, we continuously “turned off” the multiplier system v2k η on PSL(2,Z)
by letting the weight k → 0 and studied the varying distribution of eigenvalues,
Nk(T ), as well as the formation of the continuous spectrum.
• There are some cases where arithmeticity plays a role even in nonzero
weight. We studied the Shimura correspondence between weight zero
forms on Γ0(2) and weight one half forms on Γ0(4). And we also studied
weight one forms on PSL(2,Z) which correspond to Hecke eigenforms
on Γ0(144) with a Dirichlet character.
Further data than that which is presented in this section is available from the
author upon request.
2.9.1 Varying weight
The first experiment considered was to tabulate the first few eigenvalues (up
to R = 14) for PSL(2,Z) and the multiplier system v2k η , of weight k ∈ (0,6]
(cf. Section 2.4.5). We made the computations for k ∈ [0.1, 6] with a grid size
of 0.01, and the results are presented in Figure 2.1. We stress here, that the
arcs in Figure 2.1 terminate at k = 0.1; it is not excluded that they might go
lower.
For some examples of eigenvalues for “large” weights see Table 2.1. This
data should be compared with data obtained by Mühlenbruch, [70, p. 160],
who used a completely different method (with much less accuracy). We note
here that as R increases, the negative Fourier coefficients seem to grow rapidly
89

Figure 2.1: Section of eigenvalues with 0 < R ≤ 14, and weight 0.1 ≤ k ≤ 6.
R
14
12
10
8
6
4
2
0
0.1 1 2 3 4 5 6
Weight k
in magnitude (as compared to the positive ones, with the normalization c(1) =
1) for “large” weights.
We believe that we have found all eigenvalues with (R,k) ∈ [0,14]×[0.1, 6].
This belief is founded on the “continuity” of the resulting graphs R j(k) (cf. Figure
2.1), where R j(k) is the j-th eigenvalue at weight k, considered as a function
of k. By standard results (e.g. [17, p. 149]) R j(k) should be a real analytic
function in this interval.
Remember that, for k ≥ 0, the smallest eigenvalue, λmin, corresponds to
the function F(z) = y k 2 η(z) 2k . A lower bound for the second smallest eigenvalue,
λ0(k), is discussed in [17, p. 183]. Bruggeman finds two such bounds,
both positive, which he calls µ0(k) and µ1(k) (µ1 is better than µ0 in a certain
interval I ⊂ [0,2].) Figure 2.2 shows a comparison between the R-values corresponding
to these bounds and the smallest experimentally found eigenvalues
in the interval k ∈ [0.1, 6]; we see that Bruggeman’s bounds can be improved
quite a bit.
2.9.2 Small weights
The investigation of eigenvalues for small weights k has been done in the
interval R ∈ [0,20], and k = 10 − j , where 1 ≤ j ≤ 12, i.e. k ∈ [10 −12 ,0.1]. We
believe that most cusp forms were found.
Let us use the notation
λ j(k)
for the j-th discrete eigenvalue at weight k, and φ j(k) for the corresponding
cusp form. It is then a basic fact that λ j(k) depends continuously on k, but
it can also be shown that for k ∈ (0,12) λ j is even real analytic in k. That is,
φ j(k) belong to an “analytic family” in the terminology of Bruggeman (see
90

Figure 2.2: Comparison with the theoretical lower bounds in k ∈ [0.1,6].
R
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
µ1(k)
µ0(k)
0.0
0.1 1 2 3 4 5 6
Weight k
[17, 18]). In connection with this, it should also be noted that our experiments
support the statement in Observation 71 in [70] that the first few cusp forms
at weight 0 do not belong to an analytic family of cusp forms defined in the
interval (−12,12). Indeed we find that in the range considered we actually
seem to have λ j(k) → 0 as k → 0 which is consistent with Bruggeman’s result
(cf. case ii) b) in Prop. 2.17 in [17, p. 149]).
Our experiments indicate that for fixed small k, the “generic” cusp forms
φ j(k) can be divided into two classes:
• C(k), and
• E(k).
Here C(k) consists of functions φ j(k) such that λ j(k) is close to an eigenvalue
of a cusp form at weight 0 and the Fourier coefficients are close to the
corresponding coefficients of the weight-zero cusp form.
E(k) on the other hand consists of functions φ j(k) such that the Fourier
coefficients are close to the Fourier coefficients of the Eisenstein series E(z,s)
where λ j(k) = s(1 − s).
The typical difference between the Fourier coefficients at weight k and
weight 0 are in both cases basically of order k; for the forms in C(k), the
distance between λ j(0) and the corresponding discrete eigenvalue at weight 0
is much smaller than k.
The “generic” in the statement above means that we exclude certain places
where the families φ j(k) changes character between C(k) and E(k). In these
cases we have a situation of almost multiplicity 2, and in too coarse resolution
it actually looks like the analytic families intersect.
Note that Weyl’s law for non-trivial η-multiplier and a fixed non zero weight
91

k on PSL(2,Z) is
T 2 T
Nk(T ) = ♯{R ≤ T } = − ln1
− e
12 π
kπ 6 i
+ S(T ) + O(1),
for T ≥ 1 (cf. [39, p. 466]). One would like to say that the last two terms are
bounded uniformly in k, and indeed one can easily show that the O(1) term
is uniformly bounded in k. It seems plausible that, using techniques from
[41, 38, 39], one should be able to prove that S(T ) is also uniformly bounded
in k as k → 0. Details in that direction might appear in a later paper, but
for now we simply make an assumption (heuristic, but backed by numerical
evidence) that S(T ) is uniformly bounded in k. We then have the following
(conjecture):
Nk(T ) = N0(T ) − T
π ln
1 − e kπ 6 i
+ O(1) = N0(T ) − T
π ln
kπ
+ O(1),
6
(2.22)
as k → 0, where
T 2
N0(T ) =
12 .
We have computed Nk(T ) experimentally for k = 10− j , j = 4,...,12, and Figure
2.3 shows a picture of the experimental values compared to the theoretical
values obtained by (2.22) (with the O(1) term neglected) and the difference
indeed seems to be constant over this range.
From the form of the Weyl law above, we also see that the successive spac-
π
ings ∆n(k) = Rn+1(k)−Rn(k) should look about like
|ln kπ 6 | , i.e. ∆n(k) ≈ π
|ln kπ 6 | .
Figure 2.4 provides a nice illustration of this fact, where it is clearly seen that
the average spacings are almost constant for small k and this constant is pro-
. The mean gap over 10 typical cases (at R ≈ 10) turned out
portional to 1
|lnk|
to be 0.118 at k = 10−9 , 0.109 at k = 10−10 and 0.100 at k = 10−11 . The rel-
ative quotients, ∆n(k)
kπ ln , 6 are 2.52, 2.58, 2.60, respectively, and it is not
inconceivable that one obtains 3.14 in the (logarithmic) limit.
Level repulsion
From figures like 2.4 it one may be tempted to think that there are horizontal
lines corresponding to cusp forms at weight 0 which crosses the lines that are
going down (i.e. corresponding to the Eisenstein series at weight 0). This is
not the case! If we look closer we will see that there is actually “level repulsion”
here, i.e. the horizontal “cusp-form-line” turns down at the crossing and
becomes an “Eisenstein-series-line”, and the previous “Eisenstein-series-line”
turns into a “cusp-form-line”. More precisely formulated: if there is a “near
crossing” at the weight k0 close to the eigenvalue R0 ≈ R j(k0) ≈ R j+1(k0), then
92

Figure 2.3: Plot of Weyl’s law with constant T = 20 and weight k → 0
Nk(20)
200
180
160
140
120
100
80
60
3 4 5 6 7 8 9 10 11 12
−log10(k)
Nk(20)
Experimental data
Figure 2.4: Section of eigenvalues with 9 ≤ R ≤ 14, and 1E − 9 ≤ k ≤ 1E − 7.
14
13
12
11
10
9
0 1e-08 2e-08 3e-08 4e-08 5e-08 6e-08 7e-08 8e-08 9e-08 1e-07
there are two analytic families φ j(k) and φ j+1(k) such that for some δ > ε > 0:
φ j(k) ∈
C(k),
E(k),
k ∈ [k0 + ε,k0 + δ],
k ∈ [k0 − δ,k0 − ε],
and
φ j+1(k) ∈
E(k), k ∈ [k0 + ε,k0 + δ],
C(k), k ∈ [k0 − δ,k0 − ε],
and in the interval (k0 − ε,k0 + ε) both families display a mixing between the
two types E(k) and C(k). In fact, the Fourier coefficients of φ j+1 converge
93

(as k → k0 − ε) to values close to the Fourier coefficients of φ j for k > k0 +
ε and vice versa. Since the two functions also need to be orthogonal it is
clear that the Fourier coefficients exhibit “wild” behaviour in the small interval
surrounding the “near crossing”. Note also that, as k → 0, all φ j(k) ∈ E(k) and
R j(k) → 0.
See Table 2.9 for examples of Fourier coefficients corresponding to eigenfunctions
of types E(k) and C(k), close to an avoided crossing at weight
k=9.044605824E-8. Table 2.10 illustrates the agreement between the Fourier
coefficients of a more generic cusp form in E(k) and the corresponding coefficients
of the Eisenstein series (appropriately normalized) at weight 0. The
level of agreement is striking to put it mildly; likewise in Table 2.9 for the
C(k) eigenfunction. The “1 for 1” nature of this convergence in the presence
of a limiting continuous spectrum seems not to have been suspected earlier.
Cf. [41, thm. 6.6 and cor. 6.9] and [40].
The fact that the system seems to avoid accidental degeneracies by means
of level repulsion and avoided crossings is in agreement with the Wigner-von
Neumann theorem, cf. [101].
2.9.3 Lifts at weight 1
As we saw in Section 2.4.7 we could prove the existence of certain Hecke
relations at weight 1 (e.g. (2.16), (2.17) and 2.18)). Tables 2.2 and 2.3 contain
numerical verifications of these relations. Table 2.4 contains a list of computed
eigenvalues on M Γ0(1),v 2 η,1,λ , and the eigenvalues corresponding
to cosine CM-forms are indicated. In these cases, we have computed the actual
error since we know the exact eigenvalues:
Rk =
2πk
ln 7 + 2 √ , k ∈ Z
12
+ .
Note that the actual error is in general much smaller than the error-parameter
which is basically H(Y1,Y2) = |c(2) − c ′ (2)| + |c(3) − c ′ (3)| + |c(4) − c ′ (4)|,
where c(n) is computed with Y1 and c ′ (n) with Y2.
2.9.4 Half integer weight
We now consider the case of Γ0(4) and the θ-multiplier system. The aim of
our investigation in this case was to study the Shimura lift, and in particular to
investigate the dimensions of the spaces of half integer weight forms.
Our numerical results indicate that the following conjectures are true:
• If M(Γ0(2),0,2R) is an even (cosine) oldspace, then M(Γ0(4),vθ , 1
2 ,R) has
dimension 2. (Cf. our earlier remark about this in §2.7.1.)
94

Figure 2.5: Level repulsion at R = 13.779...
R
R
R
14.1
14.0
13.9
13.8
13.7
13.6
13.5
1.0e-03 1.2e-03 1.4e-03 1.6e-03 1.8e-03
13.7805
13.7800
13.7795
13.779760
13.779755
13.779750
(a)
Weight k
1.6360e-03 1.6366e-03 1.6372e-03 1.6378e-03
(c)
Weight k
1.636794e-03 1.636800e-03 1.636806e-03
(e)
Weight k
R
R
R
13.820
13.800
13.780
13.760
13.77980
13.77975
13.77970
13.7797515
13.7797510
13.7797505
1.60e-03 1.63e-03 1.66e-03 1.69e-03
(b)
Weight k
1.63670e-03 1.63676e-03 1.63682e-03 1.63688e-03
(d)
Weight k
1.6367962e-03 1.6367970e-03 1.6367978e-03
(f)
Weight k
• In such 2-dimensional space, we can choose one (basis) function from the
space V + , i.e. it has c1(n) = 0 if n ≡ 2,3 mod 4 (cf. §2.7.1).
• Suppose that f ∈ M(Γ0(2),0,2R) is an even (cosine) newform, which satisfy
f − 1
2z = ε f (z), where ε = ±1. Then M(Γ0(4),vθ , 1
2 ,R) is one dimensional.
Furthermore, if the inverse Shimura lift of f , φ ∈ M(Γ0(4),vθ , 1
2 ,R)
has Fourier coefficients c(n) (with respect to ∞) then there are two cases:
• If ε = 1 we can normalize φ with c(n) = 1 and we also have c(n) = 0
for n ≡ 5 mod 8 (cf. R = 4.4614... in Table 2.9.4).
• If ε = −1 then c(1) = 0, and we can normalize φ with c(2) = 1 and
we also have c(n 2 ) = 0 for all n and c(n) = 0 for n ≡ 1 mod 8 (cf. R =
6.0464... in Table 2.9.4).
95

Figure 2.6: Level repulsion at R = 9.533...
R
R
R
10.0
9.8
9.6
9.4
9.2
9.0
9.533700
9.533695
9.533690
9.53369535
9.53369530
9.53369525
9.53369520
9.53369515
1.20e-08 1.60e-08 2.00e-08 2.40e-08 2.80e-08
(a)
Weight k
2.43549e-08 2.43551e-08 2.43553e-08 2.43555e-08
(c)
Weight k
2.4355247e-08 2.4355252e-08 2.4355257e-08
(e)
Weight k
R
R
R
9.53378
9.53374
9.53370
9.53366
9.53362
9.5336956
9.5336954
9.5336952
9.5336950
9.533695264
9.533695262
9.533695260
9.533695258
9.533695256
2.4350e-08 2.4353e-08 2.4356e-08 2.4359e-08
(b)
Weight k
2.435523e-08 2.435525e-08 2.435526e-08 2.435528e-08
(d)
Weight k
2.43552545e-08 2.43552547e-08 2.43552549e-08
(f)
Weight k
Tables 2.5 and 2.9.4 contain examples of Fourier coefficients at weight 1
2 and
Table 2.8 contains a comparison of Fourier coefficients computed both from
forms on Γ0(2) via (2.20) and computed directly. A supplement of Fourier
coefficients for the weight 0 forms is available in Table 2.7.
96

Table 2.1: Eigenvalues for N = 1, v = vη
k = 5.0
R |c(−1)| a H(y1,y2)
3.66240686691 2E+3 8E-09
5.77698688079 6E+3 1E-09
6.64285171609 1E+4 2E-09
7.82634704661 7E+4 8E-07
8.66620831839 8E+4 1E-08
9.45156176778 4E+4 4E-09
10.21802876612 9E+4 2E-08
10.65897262925 2E+5 5E-09
11.27526358329 2E+5 2E-08
12.15792337439 5E+6 2E-06
12.55403510011 3E+5 3E-09
13.00123950671 4E+4 2E-09
13.67542640619 8E+5 8E-09
13.71353384347 4E+6 4E-07
14.47039277248 6E+5 1E-09
15.03845367363 1E+6 6E-09
15.39856858318 1E+6 1E-09
15.85705128856 7E+4 7E-09
16.14536205683 5E+6 2E-07
16.45061260141 2E+6 1E-08
16.93043847901 2E+7 2E-08
17.51562192888 2E+5 4E-09
17.59022138300 1E+6 6E-10
18.13826107361 7E+6 2E-08
18.32637702289 5E+5 5E-09
18.76341585136 2E+6 5E-09
19.16629116326 4E+6 8E-10
19.67214438521 3E+7 2E-07
19.68520099819 1E+6 3E-10
20.00524829746 2E+6 4E-09
20.38266630653 3E+6 3E-10
20.67297062056 6E+6 4E-08
20.97339376061 6E+6 8E-10
k = 5.25
a The normalization we have used here is the usual c(1) = 1.
R |c(−1)| a H(y1,y2)
3.68037312372 3E+3 3E-08
5.82067054942 9E+3 6E-09
6.63460520751 2E+4 3E-09
7.90867228426 2E+5 9E-09
8.61646891946 1E+5 8E-09
9.56930344151 7E+4 5E-09
10.15656706121 2E+5 2E-09
10.70911890024 2E+5 4E-10
11.34046324165 4E+5 2E-08
12.11839521329 4E+6 2E-06
12.65021958486 4E+5 9E-09
13.02622821839 8E+4 2E-09
13.56022943627 1E+6 1E-07
13.87057635696 7E+5 4E-07
14.48204838116 1E+6 7E-05
15.09966704087 4E+7 1E-07
15.38981845044 1E+6 4E-09
15.94059443942 1E+4 3E-09
16.09999759486 8E+6 7E-06
16.52671557073 5E+6 1E-09
16.90856097808 2E+7 5E-08
17.53730159778 1E+6 3E-10
17.74142373355 7E+6 4E-09
18.02022951826 6E+6 9E-10
18.37970066644 8E+5 2E-09
18.90587158951 1E+7 8E-09
19.09726131554 1E+7 5E-10
19.66894569593 5E+6 3E-10
19.73195996101 7E+6 4E-10
20.12609436572 2E+6 3E-10
20.35571778301 1E+7 7E-10
20.71020380483 6E+6 8E-09
20.88321504381 1E+7 2E-09
97

Table 2.2: Fourier coefficients for a CM-form
f ∈ M(Γ0(1),v 2 η,1,4.770984191561)
n c(n)/c(0) a Error
0 1.755930576575
1 1.000000000000
2 −1.755930576574 c(27) 0.4E-08
3 1.571810322167 c(40) 0.1E-08
4 1.755930576575 c(53) 0.7E-08
5 −1.770268323978 c(66) 0.3E-09
6 −2.474798320759 c(79) 0.1E-08
7 0.000000000000 c(92) 0.4E-08
8 0.346240855507 c(105) 0.5E-08
9 3.510179255561 c(118) 0.3E-08
10 1.116593241680 c(131) 0.4E-08
11 −0.000000000001 c(144) 0.3E-08
12 0.000000000001 c(157) 0.1E-07
13 −3.019229958496 c(170) 0.8E-08
14 −1.186431979458 c(183) + c(1) 0.3E-08
15 −3.079783541463 c(196) 0.5E-08
−c(n)c(−1)/R 2 /c(0)
-1 5.055064268188 b 0.1E-11
-2 −11.180067729976 c(21) 0.4E-07
-3 0.000000000001 c(32) 0.2E-07
-4 16.472675660354 c(43) 0.2E-08
-5 16.729030199659 c(54) 0.5E-07
-6 13.098490835617 c(65) 0.5E-08
-7 16.046414105740 c(76) 0.1E-07
-8 −0.000000000023 c(87) 0.1E-07
-9 13.340740291248 c(98) 0.3E-07
-10 0.000000000006 c(109) 0.4E-07
-11 2.151215034503 c(120) 0.7E-08
-12 2.878852009050 c(131) − c(1) 0.2E-07
-13 0.000000000039 c(142) 0.6E-08
-14 7.496795049955 c(153) 0.4E-07
-15 −12.830881007618 c(164) 0.4E-07
a This quotient is deduced from formula (2.14) or (2.15) on p.
79.
b c(−1) 2 = −R2c(0)(c(10) − c(0))
98

Table 2.3: Fourier coefficients for a non-CM-form
f ∈ M(Γ0(1),v 2 η,1,3.66240686698667)
n c(n)/c(0) a Error
0 −1.352193685534
1 1.000000000000
2 −1.697113317091 c(27) 0.5E-08
3 −0.057989599353 c(40) 0.4E-07
4 2.461764397786 c(53) 0.1E-07
5 0.510856433057 c(66) 0.6E-08
6 −0.952325903762 c(79) 0.8E-08
7 −1.660630683908 c(92) 0.2E-07
8 −2.343382246022 c(105) 0.9E-08
9 1.271097206907 c(118) 0.1E-07
10 −0.203512820511 c(131) 0.6E-08
11 2.110622602834 c(144) 0.5E-08
12 2.170616908700 c(157) 0.4E-08
13 0.449799127363 c(170) 0.4E-07
14 0.612654661780 c(183) + c(1) 0.4E-07
15 −1.684453740441 c(196) 0.1E-07
16 0.400312170289 c(209) 0.2E-07
17 −2.868395110060 c(222) 0.1E-07
18 −1.931595991172 c(235) 0.2E-07
19 −0.591212766919 c(248) 0.1E-07
20 0.792151138999 c(261) 0.9E-08
21 −1.717242193922 c(274) 0.3E-07
22 1.369169138277 c(287) 0.2E-07
23 2.007854712832 c(300) 0.8E-09
24 0.447826147902 c(313) 0.1E-07
25 3.051006373828 c(326) 0.4E-08
26 −0.032419986064 c(339) 0.6E-08
27 1.255081531820 c(352) + a(2) 0.2E-07
28 −0.707047087424 c(365) 0.3E-07
29 1.272283355260 c(378) 0.1E-07
30 −0.184187214400 c(391) 0.2E-07
a This quotient is deduced from formula (2.14) or
(2.15) on p. 79.
99

Table 2.4: Eigenvalues for M(PSL(2,Z),1,v 2 η)
R H(y1,y2) True error a
2.38549209578045 1E-12 1E-15 b
3.66240686698667 1E-11
4.77098419156091 1E-13 1E-14 b
5.77698688078694 6E-12
6.64285171613711 1E-11
7.15647628734173 1E-11 4E-13 b
7.82634704540775 1E-13
8.66620831896793 7E-14
9.45156176783224 7E-12
9.54196838312186 2E-13 6E-14 b
10.21802876776059 3E-13
10.65897262920241 2E-12
11.27526358349387 2E-13
11.92746047890219 7E-11 5E-14 b
12.15792337422149 7E-13
12.55403509998720 1E-13
13.00123950642372 9E-13
13.67542640643589 2E-13
13.71353384358095 1E-12
14.31295257468268 1E-12 1E-14 b
14.47039277253940 1E-12
15.03845367358721 2E-13
15.39856858348441 1E-12
15.85705128717333 3E-10
16.14536205734475 3E-11
16.45061260131967 4E-11
16.69844467046304 4E-12 1E-13 a
16.93043847896222 4E-15
17.51562192885174 2E-10
17.59022138305996 1E-10
18.13826107340244 3E-11
18.32637702205910 4E-11
18.76341585146817 1E-11
a For CM-forms, the true error is computed with
respect to the eigenvalue Rk = 2πk
ln(η0) , where
η0 = 7 + 2 √ 12.
b These forms correspond to CM-forms.
100

Table 2.5: Fourier coefficients for
f1,2 ∈ M(Γ0(4), 1
2 ,6.889875675945)
Fourier coefficients for f1 ∈ V + , observe that a(n) = 0 for n ≡ 2,3 mod 4
n a(n)
4 0.84219769675471
5 0.18355821406443
8 0.56907998524429
9 −0.33045049673565
12 −0.41296169213831
13 0.60153537988057
16 0.30482066289397
17 −0.88689598690620
20 0.41418282092059
21 0.50212917023175
23 −0.00000000000110
24 −1.04429548341249
25 0.28984678984391
n a(−n)
|c(4)c(9) − c(36)| = 0.2E − 08
3 1.01825299171456
4 2.18968040385979
7 2.06305218095270
8 −1.17157116610978
11 −1.02718719694121
12 2.29759751514531
15 −4.08935474990375
16 3.39248165496032
19 −1.31804633824673
20 1.87725570455517
23 −2.22265197818114
24 0.58816620330140
Fourier coefficients for f2 ∈ V + , observe that a(1) = 0, and we set c(2) = 1
n a(n) n a(n)
0 0.000000000000 19 −0.936283350934
1 0.000000000000 20 0.192087703124
2 1.000000000000 21 1.247834928335
3 −0.725665465042 22 0.411443268212
4 −0.710754741008 23 −0.018564690418
5 0.456158224759 24 −1.297582830232
6 −1.835059236817 25 0.000000000000
7 0.481289183972 26 −0.610656711780
8 0.707106781187 27 −0.179166638197
9 0.000000000000 28 0.340322845698
10 −0.651049038069 29 0.001563849679
11 −0.470914040036 30 1.462745418892
12 −0.513122971204 31 0.400763051265
13 1.494868058150 32 0.095523702475
14 0.968484734380 33 0.000000000000
15 −0.945563574521 34 −0.879782299796
16 −1.101175502961 35 −1.833603623066
17 0.000000000000 36 0.234869257223
18 0.824250041644
101

Table 2.6: Fourier coefficients for f ∈ M Γ0(4), 1
2 ,R
102
R = 4.461438243496
n a(n)
0 0.00000000000000
1 1.00000000000000
2 0.63334968449036
3 0.63517832947402
4 −0.70710678118667
5 −0.00000000000003
6 1.28035706400142
7 −0.90756258916698
8 −0.44784585676555
9 0.52643872643776
10 −0.57763498966972
11 1.12485377915641
12 −0.44913890403389
13 0.00000000000001
14 0.48078071833327
15 −1.58539012005784
16 0.50000000000015
17 −0.14882069214483
18 1.06474902295605
19 0.21268916863632
20 0.00000000000003
21 0.00000000000002
22 −1.56248056455209
23 0.85478501960318
24 −0.90534916229569
25 0.45696099733973
36 −0.37224839333969
|c(4)c(9) − c(36)| = 4E − 12
R = 6.046497437542
n a(n)
0 0.000000000000
1 0.000000000000
2 1.000000000000
3 1.770795863371
4 0.000000000029
5 1.331470494003
6 −0.749395507674
7 0.074369313625
8 0.707106781209
9 −0.000000000016
10 −1.214451367832
11 0.620756243833
12 1.252141763204
13 −1.123686021586
14 −0.711286031630
15 0.964706032413
16 −0.000000000112
17 −0.000000000108
18 −0.128644484609
19 −0.499664871594
20 0.941491814843
21 −0.447011950327
22 −0.805501416928
23 1.236716356721
24 −0.529902643583
25 0.000000008756
36 −0.000000000168
288 −0.064322242377
|c(2 · 3 2 )c(2 · 4 2 ) − c(2 · 12 2 )| = 8E − 11

Table 2.7: Supplemental table of Fourier coefficients for M(Γ0(2),0,R)
R 8.92287648699174
n A(n)
2 −0.70710678118654
3 1.10378899562734
4 0.49999999999993
5 0.90417459283958
6 −0.78049668380711
7 0.82934246755499
8 −0.35355339059330
9 0.21835014686776
10 −0.63934798597339
R 13.77975135189073
(even wrt z ↦→ − 1 2z )
n A(n)
2 2.96351804031448
3 0.24689977245401
4 3.59139177031902
5 0.73706038534834
6 0.73169192981688
7 −0.26142007576500
8 2.60064131148226
9 −0.93904050235826
10 2.18429174878080
R 12.0929948750786
n A(n)
2 0.70710678118655
3 −0.70599475399569
4 0.49999999999999
5 −0.79974825694039
6 −0.49921367803249
7 −1.71337067862845
8 0.35355339059328
9 −0.50157140733054
10 −0.56550741572467
R 13.77975135189073
(odd wrt z ↦→ − 1 2z )
n A(n)
2 0.13509091556820
3 0.24689977245398
4 −0.79070303958101
5 0.73706038534830
6 0.03335391631439
7 −0.26142007576538
8 −1.36013067551284
9 −0.93904050236089
10 0.09957016228575
103

Table 2.8: Comparison of Fourier coefficients A(n) computed directly for
M(Γ0(N),0,R) (N = 1,2), vs Â(n) computed on M Γ0(4), 1 1
2 , 2R and using
(2.20).
n Â(n) A(n) |A(n) − Â(n)|
f ∈ M(Γ0(2),0,8.922876486992) (t = 1)
2 −0.70710678118665 −0.707106781186 0.6E-12
3 1.10378899562739 1.103788995627 0.4E-12
5 0.90417459283969 0.904174592840 0.3E-12
f ⊆ M(Γ0(1),0,13.779751351891) (t = 1)
2 1.54930447794126 1.549304477941 0.7E-12
3 0.24689977245398 0.246899772454 0.3E-12
4 1.40034436536892 1.400344365369 0.2E-12
5 0.73706038534387 0.737060385348 0.4E-11
6 0.38252292109716 0.382522923066 0.2E-08
f ∈ M(Γ0(1),0,13.779751351891) (t = 2)
3 0.24689977245437 0.246899772454 0.8E-13
5 0.73706038535004 0.737060385348 0.2E-11
7 −0.26142007624377 −0.261420075765 0.5E-09
9 −0.93904050238904 −0.939040502362 0.3E-10
f ∈ M(Γ0(2),0,12.092994875079) (t = 2)
3 −0.70599475379863 −0.705994753996 0.2E-09
5 −0.79974825694696 −0.799748256940 0.7E-11
7 −1.71337067860377 −1.713370678628 0.2E-10
9 −0.50157140750090 −0.501571407330 0.2E-09
In segment 3, the calculation is based on (2.20) and the second portion of Table 2.5.
104

Table 2.9: Comparison of Fourier coefficients for weights k =9.044605824E-08 and
k = 0 near an “avoided crossing”.
Corresponds to the cusp form
k
R
c(2)
c(3)
c(4)
c(5)
c(6)
9.0446058240E-08
13.77975135189074
1.54930480559976
0.24689988546553
1.40034433555250
0.73706067260516
0.38252272069428
Corresponds to the Eisenstein series
k
R
c(2)
c(3)
c(4)
c(5)
c(6)
9.0446058240E-08
13.77975135138225
−2.06525760334129
−1.72891679536648
2.97153986917404
−2.02747287754385
3.41008729668221
0
13.77975135189074
1.54930447794069
0.24689977245411
1.40034436536841
0.73706038534787
0.38252292306557
0
13.77975135138225
−1.98398933080188
−1.68449330640991
2.93621366473571
−1.96531634618530
3.34201674772446
Difference
0.3E-06
0.1E-06
0.1E-06
0.2E-06
0.2E-06
Difference
0.8E-01
0.4E-01
0.4E-01
0.6E-01
0.7E-01
The coefficients for the Eisenstein series at k = 0 were computed using (2.19), i.e.:
c(2) = 2cos(Rln2)
c(3) = 2cos(Rln3)
c(4) = 1 + 2cos(Rln4)
c(5) = 2cos(Rln5)
c(6) = 2cos(Rln6) + 2cos(R(ln3 − ln2))
Table 2.10: Comparison of Fourier coefficients for weights k =9.044605824E-08 and
k = 0 “far” from an ”avoided crossing”. The weight 0 coefficients were computed
using the formulas in Table 2.9.
k 9.0446058240E-08 0
Difference
R
c(2)
c(3)
c(4)
c(5)
c(6)
13.62696884857618
−1.99957085683552
−1.48069687587703
2.99828354611637
−1.99647405201235
2.96075820067617
13.62696884857618
−1.99957081810438
−1.48069680342062
2.99828345661464
−1.99647406885962
2.96075811858031
0.4E-07
0.7E-07
0.7E-07
0.2E-07
0.8E-07
105

3 Computation of Maass waveforms on
Non-congruence Subgroups of the Modular
Group
3.1 Introduction
Previously, in Chapter 1 we considered the computation of Maass waveforms
on certain subgroups of PSL(2,Z), the so called Hecke congruence subgroups,
and in Chapter 2 we extended this work to the case of arbitrary real weight
and non-trivial multiplier system. In this chapter we will see that the methods
detailed in the previous chapters also work for non-congruence subgroups.
There is a general belief that there exist no (or very few) Maass waveforms
on a generic Fuchsian group. This is indicated by the results of Phillips and
Sarnak, cf. [80, 81]. Recent results (cf. [30]) indicate that there are continuous
families of groups, on which there exists (at least) one Maass waveform.
These families seem to be “arbitrary” in the sense that they are not arithmetic
and possess no obvious symmetries.
If a group Γ has Maass waveforms, then these descend to all subgroups of
Γ. As a Maass waveform on the subgroup we will call it an oldform. We will
use the term newform to denote the Maass waveforms on a group G which
are not from a supergroup G1 (G G1 ⊆ PSL(2,Z)). This is an analog of
the well-known concepts of oldforms and newforms for congruence groups
(cf. [5]).
The groups which we will study here are all subgroups of the modular
group, PSL(2,Z), and hence arithmetic, but even in this case the existence
or nonexistence of newforms is in general not known.
The existence of an infinite number of newforms is known on congruence
subgroups and cycloidal subgroups (having only one cusp), and other subgroups
which possess certain symmetries (e.g. a reflectional symmetry which
preserves cusp classes). The results presented in this thesis indicate that there
are no other examples of subgroups of index ≤ 11 which have newforms.
We will give a general method that produces all subgroups with a given
index in PSL(2,Z), and then we will see how to adapt the previous algorithm
for computation of Maass waveforms (cf. Section 1.3) to such a group, and
finally we will provide examples of the experimental results.
The author would also like to express gratitude to Andreas Strömbergsson
107

for very helpful discussions, especially regarding Proposition 3.2.1.
3.2 Subgroups of the Modular Group
Remember that the modular group, PSL(2,Z), is defined as
a b ad
PSL(2,Z) =
− bc = 1,a,b,c,d ∈ Z ,
c d
with a (closed) fundamental domain
F1 = z = x + iy ∈ H
1
|z| ≥ 1, |x| ≤ .
2
For a given positive integer N the principal congruence subgroup, Γ(N), is
defined by
a b
Γ(N) =
∈ PSL(2,Z)
c d
c ≡ b ≡ 0 mod N,a ≡ d ≡ 1 mod N ,
and any subgroup of PSL(2,Z) containing some Γ(N) is called a congruence
subgroup. If G ⊆ PSL(2,Z), and N is the smallest integer such that Γ(N) ⊆ G
then we say that N is the level of G. A particularly interesting and well understood
class of congruence subgroups are the Hecke congruence subgroups,
Γ0(N),
a b
Γ0(N) =
∈ PSL(2,Z)
c d
c ≡ 0 mod N .
Presentations, constructions, modular forms and spectral theory are more or
less well known and understood topics on congruence groups in general and
on Hecke congruence groups in particular, cf. e.g. [83, 24, 76, 84, 89, 51, 5,
62, 12].
Of the other types of subgroups, the cycloidal groups have been studied
for a long time by Petersson, Millington, etc. (cf., e.g. [78, 79, 66, 67]), and
other specific types of subgroups have also been studied by various people
(cf., e.g. [22, 23, 26]). The subject of interest to these investigations however
has mostly been in presenting and constructing the groups. The present
knowledge of the theory of modular forms and spectral theory on these groups
is very limited (cf., e.g. [7, 105, 106, 100]).
It is a well-known fact that PSL(2,Z) is generated by the elements S : z ↦→
z + 1, and E : z ↦→ − 1
z , and that the only relations are E2 = R 3 = Id, where
108

R = ES. In terms of matrices we have
S =
1
0
1
1
, E =
0
1
−1
0
, and R =
0
1
−1
1
.
Let Γ ⊆ PSL(2,Z) be a subgroup of finite index µ with closed fundamental
domain F in H. Suppose that F has e2 and e3 inequivalent elliptic vertices of
order 2 and 3 respectively and κ inequivalent parabolic vertices (cusps). The
genus g of the quotient orbifold, Γ\H, is given by
g = 1 + 1
2
µ 1
− κ −
6 2 e2 − 2
3 e3
, (3.1)
(cf. [67, (1.1)]) and we say that the type of Γ is (µ; g,κ,e2,e3). If µ j, j =
1,...,κ are the widths of the cusps of F then µ = µ1 +···+ µκ. (This formula
is easily verified by viewing F as µ copies of F1, the standard fundamental
domain for PSL(2,Z),since then µ j is exactly the number of images of F1
whose vertex at infinity belongs to the cusp class number j.) The generalized
level of Γ is defined as the least common multiple of µ j
κ j=1
. It is known that
if Γ is a congruence subgroup then the generalized level is equal to the level
(cf. [104]).
Theorem 3.1. (Millington [67])
For every set of integers µ > 0, κ > 0, e2 ≥ 0, e3 ≥ 0, g ≥ 0 consistent with
(3.1) there exists a subgroup of the modular group of type
(µ; g,κ,e2,e3).
3.2.1 Permutations and subgroups of PSL(2,Z)
There is a convenient way to represent any subgroup G of PSL(2,Z) of finite
index µ in terms of a pair of permutations of a set of µ elements. Geometrically
these permutations can be viewed as describing the side pair identifications
in a fundamental domain for G formed by µ distinct PSL(2,Z)-translates
of F1 (cf. Remark 3.2.1 for a precise description of this).
For a positive integer µ we use Sµ to denote the group of permutations of
µ-letters (e.g. {1,2,..., µ}). The notation here is that the action of a permutation
π ∈ Sµ is from the left, i.e. for π,σ ∈ Sµ we write πσ( j) = π (σ( j)).
A k−cycle is a permutation π ∈ Sµ with the property that there exist k distinct
elements x1,x2,...,xk ∈ {1,2,..., µ} such that π(xi) = xi+1, i = 1,...,k−
1 and π(xk) = x1. We will use the following notation: π = (x1x2 ···xk). Note
that with this notation, (x1x2 ···xk) = (x2x3 ···xkx1) = ... .
A general permutation π is completely described by either listing its values
109

on all elements, i.e. {π( j)} µ
j=1 or by listing all cycles, i.e.
with the sets x j1,xj2,...,xjk j
π = (x11 ···x1k1 )(x21 ···x2k2 )···(xn1 ···xnkn ),
( j = 1,2,...,n) pairwise disjoint.
Definition 3.2.1. If σE,σR ∈ Sµ we say that the pair (σE,σR) is legitimate if
• σ 2 E = σ 3 R = 1, and
• the group generated by σE and σR, 〈σE,σR〉, is transitive on {1,2,..., µ}.
Two legitimate pairs (σE,σR) and (σ ′ E ,σ′ R ) are said to be equivalent, denoted
by (σE,σR) ∼ (σ ′ E ,σ′ R ), if there exists a permutation π ∈ Sµ such that
π −1 σEπ = σ ′ E, and π −1 σRπ = σ ′ R.
If we also have π(1) = 1, we say that the pairs are equivalent (mod 1), which
we denote by (σE,σR) ∼1 (σ ′ E ,σ′ R ).
We have the following theorem.
Theorem 3.2. (Millington [67, thm. 2])
There is a one-to-one correspondence between equivalence classes (mod 1)
(i.e. with respect to ∼1) of legitimate pairs (σE,σR) in Sµ and subgroups G
of index µ in PSL(2,Z). Furthermore, G has type (µ; g,κ,e2,e3) and cusp
widths µ j
κ j=1
if and only if
• σE has e2 1−cycles (fixed elements),
• σR has e3 1−cycles (fixed elements), and
• σS = σRσE has κ disjoint cycles of lengths µ j, 1 ≤ j ≤ κ
(and g is given by (3.1)).
Remark 3.2.1. The subgroup, G, of PSL(2,Z) corresponding to the legitimate
pair (σE,σR) is defined by
G = {γ ∈ PSL(2,Z)|h(γ)(1) = 1},
where h : PSL(2,Z) → Sµ is the anti-homomorphism (i.e. h(AB) = h(B)h(A),
for all A,B ∈ PSL(2,Z) and h(Id) = Id) defined by h(E) = σE and h(R) = σR.
In the implementation of the general pullback algorithm (cf. Section 1.3.2) we
use the fact that a set of right coset representatives of G in PSL(2,Z) is given
by µ
Vj , where
j=1
h(Vj)(1) = j, j = 1,..., µ.
Hence a fundamental domain for G\H is given by F = ⊔ µ
j=1Vj (F1). One
checks from the definitions that, for all j, h
(1) = 1 so that for all
VjSV −1
h(S)( j)
j, VjSV −1
h(S)( j) ∈ G. This means that VjS(F1) can be mapped to Vh(S)( j) (F1) by
a map in G. Let b be the geodesic ei π 3 + i[0,∞) so that b and S−1 (b) are two
110

of the three sides of the geodesic polygon F1. Since Vj (F1) ∩VjS(F1) = Vj(b)
it follows that there is an identification map in G which glues the side Vj(b)
of Vj (F1) to the side V h(S)( j)S −1 (b) of V h(S)( j) (F1). Similarly, if b ′ denotes
the side of the polygon F1 which contains the point i, then the side Vj(b ′ ) of
Vj (F1) is glued to the side V h(E)( j)(b ′ ) (provided with the “opposite direction”)
of V h(E)( j) (F1).
For the pullback procedure, we need to find the coset of the map T −1 , where
T gives the pullback into F1, i.e. we need to find h(T −1 )(1). To do that, as
we go through with the “flip-flop” through the generators E and S, in each
step we both multiply the appropriate map (from the left) and the inverse of the
corresponding permutation (from the left). When we are done, we have both T
and h(T −1 ) and can thus determine the coset of T −1 . (The G-pullback is then
given by the map V h(T −1 )(1)T , cf. Chapter 1, p. 43.)
(In her thesis, Millington appears to get a connected polygon with identified
sides by linking the permutations with the Poincaré fundamental polygon theorem
in an appropriate way. See [66, proof of thm. 1] for the cycloidal case.
In our approach, we do not need a connected polygon and simply proceed
algebraically by way of [67, thm. 1].)
Theorem 3.3. Let (σE,σR) correspond to G and (σ ′ E ,σ′ R ) to G′ . Then G and
G ′ are conjugate in PSL(2,Z) iff (σE,σR) ∼ (σ ′ E ,σ′ R ).
Proof. Let h and h ′ denote the anti-homomorphisms corresponding to (σE,σR)
and (σ ′ E ,σ′ R ). Suppose that A ∈ SL(2,Z). We then know that
AGA −1 = A γ ∈ PSL(2,Z) h(γ)(1) = 1 A −1
= AγA −1 ∈ PSL(2,Z) h(γ)(1) = 1
= γ ∈ PSL(2,Z) h(A)h(γ)h(A) −1 (1) = 1
= γ ∈ PSL(2,Z) h ′′ (γ)(1) = 1 ,
where h ′′ (γ) = h(A)h(γ)h(A) −1 ∈ Sµ corresponds to the legitimate pair (σ ′′
E ,σ′′ R )
(here σ ′′
E = h(A)σEh(A) −1 and σ ′′
R = h(A)σRh(A) −1 ). Suppose that G ′ = AGA−1 .
This is equivalent with
γ ∈ PSL(2,Z) h ′′
(γ)(1) = 1 = γ ∈ PSL(2,Z) h ′
(γ)(1) = 1 ,
), which
means that there exists a π ∈ Sµ with π(1) = 1 such that π−1 (σ ′ E ,σ′ R )π =
and by theorem 3.2 we must have then have (σ ′′
E ,σ′′
R ) ∼1 (σ ′ E ,σ′ R
(σ ′′
E ,σ′′
R ) and if we set τ = h(A)−1 π −1 we have τ −1 (σE,σR)τ = (σ ′ E ,σ′ R )
where now τ(1) = h(A) −1 (1). Hence (σE,σR) ∼ (σ ′ E ,σ′ R ) .
Conversely, suppose that τ ∈ Sµ and τ−1 (σE,σR)τ = (σ ′ E ,σ′ R ). Then, for
any A ∈ SL(2,Z) such that h(A) −1 (1) = τ(1), we have
G ′ = AGA −1 .
111

This is easily seen by comparing h ′ (γ) = τ −1 h(γ)τ and h ′′ (γ) = h(A)h(γ)h(A) −1
for γ ∈ PSL(2,Z).
Remark 3.2.2. If G ′ = AGA−1 , A ∈ SL(2,Z) then if j = h(A) −1 (1) and σ j =
(1 j), the pair (σ ′ E ,σ′ R ) defined by
′
σ E,σ ′ −1
R = σ j (σE,σR)σj
is easily seen to correspond to the group G ′ . Hence we see that all conjugates
of G are obtained by conjugating the corresponding pair (σE,σR) by σ j for
j = 2,..., µ.
Definition 3.2.2. If G is a subgroup of PSL(2,Z) then we use (G) to denote
the set of PSL(2,Z)-conjugates of G, (G) = AGA −1 A ∈ PSL(2,Z) .
It is clear now that if (σE,σR) is a legitimate pair corresponding to G, then
we get all groups in (G) by conjugating the pair (σE,σR) with all σ j = (1 j),
j = 2,..., µ.
Remark 3.2.3. We know that conjugate groups have the same spectrum, hence
we only need to investigate one G in the set (G).
The reflection operator J : z ↦→ −z, which we also represent by the matrix
J =
−1
0
0
1
,
acts on E and R as
E ∗ = JEJ = E, and R ∗ = JRJ = ER −1 E = ER 2 E.
Hence, if G is represented by the legitimate pair (σE,σR), the group G∗ = JGJ
corresponds to the legitimate pair
.
(σ ∗ E,σ ∗ R) = σE,σEσ 2 RσE
We also observe that G ∗ has the same type as G and the spectrum of G and
G ∗ are equal. In fact, it is obvious that if f is a function on G\H, then f |J is a
function on G ∗ \H.
Remark 3.2.4. Since one can easily compute the pair corresponding to G ∗
from the pair corresponding to G we do not list (G ∗ ) separately in Table 3.1.
But for each conjugacy class (G) we list the conjugacy class of (G ∗ ). Clearly
either (G ∗ ) = (G) or (G ∗ ) = (G). If (G ∗ ) = (G) we can in some cases use
the reflection to prove existence of newforms (cf Proposition 3.2.1 and Remark
3.2.1).
112

We will now give a precise statement about a sufficient condition that newforms
exist on a subgroup of PSL(2,Z). This proposition gives an explanation
for all newforms on noncongruence subgroups which we have found so far.
Proposition 3.2.1. Suppose that G corresponds to the legitimate pair (σE,σR),
and that JGJ = A −1 GA for some A ∈ SL(2,Z). Let P = {[p1],...,[pκ]} be the
set of G-equivalence classes of cusps of G. Set α = AJ and assume that α 2 ∈ G
and that α preserves all cusp classes. Then L 2 (G\H) splits into an “α-even”
and “α-odd” part, and the α-odd part is spanned by Maass waveforms.
Proof. Extending [94, I, §§1-2] to PGL(2,R) (cf. Chapter 1, p. 21) we see that
α gives us a modular correspondence (with d = 1). Since the cusp classes are
preserved by α we can use [94, lemma 2.6] (extended to PGL(2,R)) to show
that there exist positive constants γ1,...,γκ such that
E j (αz,s) ≡ γ s jEj (z,s), for j = 1,...,κ, (3.2)
where E j (z,s) is the Eisenstein series for G\H associated with [p j]. Since
α2
∈ G we have E j α2z,s ≡ E j (z,s), and hence, in fact γ1 = γ2 = ··· = γκ =
1. L2 (G\H) splits into an “even” and an “odd” part by the action of α;
L 2 (G\H) = L 2 + (G\H) ⊕ L 2 − (G\H),
(an orthogonal sum, since the action of α is self adjoint, by an argument similar
to [38, pp. 470-472] but easier) where
L 2 ± (G\H) = f ∈ L 2 (G\H) f (αz) ≡ ± f (z) .
It is known that L 2 (G\H) has a spectral decomposition as an orthogonal sum
L 2 (G\H) = C ⊕D, where C (“the continuous part”) is spanned by the Eisenstein
series, and D is spanned by Maass waveforms. By (3.2) (where γ j = 1)
we have C ⊂ L 2 + (G\H). Hence L 2 − (G\H) ⊂ D, i.e. L 2 − (G\H) is spanned
by Maass waveforms. It will be seen momentarily (cf. Remark 3.2.5) that, in
many cases, the space L 2 − (G\H) can be seen to be infinite-dimensional.
Remark 3.2.5. In each case which we have investigated and for which Prop. 3.2.1
applies, the group G does not allow any nontrivial supergroups G ′ (G G ′
PSL(2,Z)). Hence, in these cases, Prop. 3.2.1 very easily implies the existence
of newforms for G.
Indeed, let V ⊂ H be a small disk such that γV ∩V = /0 for all γ ∈ GL(2,Z);
write PSL(2,Z) = ⊔n j=1Gγ j, and let f0 be any function {1,...,n} → C with
∑ n j=1 f0( j) = 0, f0 ≡ 0. Define f : H → C by
⎧
⎪⎨ f0( j), if z ∈ Gγ jV,
f (z) = − f0( j), if z ∈ Gαγ
⎪⎩
jV,
0, otherwise.
113

Then f is G-invariant and f ∈ L 2 − (G\H) (using αG = Gα = GαG). Further-
more, if f1 ∈ L2 (PSL(2,Z)\H) then
f f1dµ =
G\H
=
=
=
= 0.
n
∑ + f f1dµ
j=1 γ jV αγ jV
n
n
∑ f0( j) f1dµ − ∑ f0( j) f1dµ
j=1 γ jV j=1 αγ jV
n
n
∑ f0( j) f1dµ − ∑ f0( j) f1dµ
j=1 V j=1 αV
n
f1dµ − f1dµ ∑ f0( j)
V
αV j=1
(We used αγ jV = αγ jα −1 αV , where αγ jα −1 ∈ PSL(2,Z)). Hence f is orthogonal
to the full oldform space (by our assumption on G). Hence, expanding f
as an L 2 -sum of Maass waveforms (using Prop. 3.2.1), all non-vanishing terms
must be newforms. And by varying V , we see that the number of newforms is
infinite.
To use the above proposition we must find a permutation τ which conjugates
the pair (σE,σR) into (σ ∗ E ,σ∗ R ). Then we must find an A ∈ PSL(2,Z) with
h(A)(1) = τ(1). And finally we set α = JA and see if α2 ∈ G, and if so, we must
also see how the various cusp classes transform under α. To be certain that
we actually find newforms we must also investigate the possible supergroups
between G and PSL(2,Z).
There are two special cases, in which we can see the exact form of α and
only have to investigate the permutation of the cusp classes and possible supergroups.
Example 3.2.1. If we can take A as a power of S : A = S k . Then α : x + iy ↦→
−x + k + iy, i.e. a reflection in the line x = k
2 and it is clear that α2 = Id ∈ G.
Example 3.2.2. If we can take A = E (i.e. we can do that if E(1) = 2 and
σR ∼1 σ 2 R , which is always true if e2 = 0 and e3 ≥ 1). Then α = EJ and
α : z ↦→ 1
z , i.e. α is a reflection in the circle |z| = 1 and α2 = Id ∈ G.
Remark 3.2.6. Consider a group G ⊆ PSL(2,Z), which has cusp width µ1 = L
at ∞, and where J acts as a reflection in the line x = k
2 , i.e. α : z ↦→ k − z.
Suppose that f is a Maass waveform on G with α-eigenvalue ε = ±1, i.e.
f (αz) = ε f (z). We know that the Fourier series of f at ∞ can be written as
nx
,
L
114
f (z) = ∑ c(n)
n=0
√ yKiR(2π|n|y)e

where e(x) = e2πix . If we apply α we get
f (αz) = ∑ c(n)
n=0
√ =
2π|n|y n(−x + k)
yKiR e
L
L
nk √yKiR
2π|n|y −nx
∑ c(n)e
e
n=0 L
L L
−nk √yKiR
2π|n|y
nx
e ,
L
L L
= ∑ c(−n)e
n=0
hence if f (αz) = ε f (z) we must have c(−n) = εc(n)e
nk
L , for all n = 0, and
in particular if we set c(1) = 1 we get
e
c(−1) =
πi
L (2k) , ε = 1, (the even case)
e πi
L (2k±L) , ε = −1, (the odd case).
We can thus determine if a given cusp form f is of even or odd type with respect
to α by looking at the c(−1) coefficient. Cf. Tables 3.5 and 3.6.
We will now consider some explicit examples where we can take A = S k as
in Example 3.2.1. See Table 3.1 for definition of these groups.
Example 3.2.3. Consider the group G = Γ10,20. One can verify (by conjugating
the corresponding permutations) that we can take A = Sk with k = 3, and
α : z ↦→ −z + 3 is a reflection in the line x = 3
2 . G has three cusps and we can
choose representatives p1 = ∞, p2 = 1 and p3 = 3
2 . Now we see that α(∞) = ∞,
α(1) = 2 and α
3 3
2 = 2 . Hence, since α2 = Id also α(p2) = p2. And thus α
preserves all cusp classes.
For any α-odd newform f on G, the Fourier coefficient c(−1) = e πi
4 (6±4) = i.
Experimentally we found only odd newforms on this group! Cf. Remark 3.2.9
and Table 3.6.
Example 3.2.4. Consider the group G = Γ10,14. One can verify that here we
can take A = S k with k = 6, and α : z ↦→ −z + 6 is a reflection in the line
x = 3. G has three cusps and we can choose representatives p1 = ∞, p2 = 1
and p3 = 3. Now we see that α(∞) = ∞, α(1) = 5 and α (3) = 3. Hence, since
α 2 = Id also α(p2) ∼ p2. And thus α preserves all cusp classes.
For any α-odd newform f on G, the Fourier coefficient c(−1) = e πi
7 (12±7) =
e 5πi
7 . Experimentally we found only odd newforms on this group! Cf. Remark
3.2.9 and Table 3.6.
Example 3.2.5. Consider the group G = Γ10,8;7. One can verify that here we
can take A = S k with k = 7. We get α : z ↦→ −z + 7. G has three cusps and
we can choose representatives p1 ∈ {∞,0}, p2 = 2 and p3 = 5. We see that
115

α(p1) = p1 and α (p2) = p3. Hence α permutes the cusps p2 and p3. And
thus α does not preserve all cusp classes. In this case, experimentally, we
found no Maass waveforms at all except for the oldforms from PSL(2,Z).
Example 3.2.6. Consider the group G = Γ10,30. One can verify that here we
can take A = S k with k = 4, and α : z ↦→ −z + 4 is a reflection in the line
x = 2. G has three cusps and we can choose representatives p1 = ∞, p2 = 1
and p3 = 2. We see that α(p1) = p1 and α(p3) = p3 and hence α (p2) = p2.
Thus α preserves all cusp classes.
For any α-odd newform f on G, the Fourier coefficient c(−1) = e πi
5 (8±5) =
e 3πi
5 . Experimentally we found only odd newforms on this group! Cf. Remark
3.2.9 and Table 3.6.
Example 3.2.7. Consider the cycloidal group G = Γ10,10,1. One can verify that
here we can take A = Sk with k = 9, and α : z ↦→ −z + 9 is a reflection in the
line x = 9
2 . G has only one cusp, ∞, and it is clearly preserved by α. Any
α-odd function f on G will have Fourier coefficient c(−1) = e πi
10 (18±10) = e 4πi
5 ,
and any α-even function will have c(−1) = e πi
10 (18) = e 9πi
5 . Experimentally we
found both odd and even newforms on this group! Cf. Remark 3.2.9 and Table
3.5.
To ascertain that we actually have newforms we have to find a way to check
for supergroups in PSL(2,Z); fortunately there is a well-known combinatorial
method to find all these. We now turn to describe this method.
Definition 3.2.3. A block system for a subgroup H ⊆ Sµ is an equivalence
relation ≡ on the set {1,2,..., µ} such that
∀h ∈ H, ∀x,y ∈ {1,2,..., µ} : x ≡ y ⇔ h(x) ≡ h(y).
An equivalence class of ≡ is called a block.
Note that for any H ⊆ Sµ there exist two trivial block systems: The block
system with µ blocks, i.e. x ≡ y for all x = y ∈,{1,2,..., µ}; and the block
system with 1 block, i.e. x ≡ y for all x,y ∈ {1,2,..., µ}.
It is easy to see that if the group H is transitive, then all blocks of H must
have the same number of elements. One can show the following proposition.
Proposition 3.2.2. Let the legitimate pair (σE,σR) correspond to the group
G ⊂ PSL(2,Z). Then there exists a supergroup G ′ , G ⊂ G ′ ⊂ PSL(2,Z) if and
only if the group 〈σE,σR〉 = h(PSL(2,Z)) ⊆ Sµ supports a non-trivial block
system.
Outline of proof. Suppose that G ′ is a supergroup of G in PSL(2,Z) and set
B = {h(γ)(1)|γ ∈ G ′ } ⊆ {1,2,..., µ}. Observe that |B| ≥ 2 if G ′ = G, and
116

|B| < µ if G ′ = PSL(2,Z). It is easy to verify that the equivalence relation
defined by
x ≡ y ⇐⇒ ∃γ ∈ PSL(2,Z) : [h(γ)(x) ∈ B and h(γ)(y) ∈ B]
gives a block system, and that [1] = B.
On the other hand, if ≡ is a non-trivial block system for h(PSL(2,Z)), then
it is easy to verify that the group
G ′ = {γ ∈ PSL(2,Z)|h(γ)(1) ≡ 1}
is a supergroup of G, G G ′ PSL(2,Z).
Remark 3.2.7. Note that an equivalence relation ≡ is a block system for the
group h(PSL(2,Z)) = 〈σE,σR〉 if and only if
∀x,y ∈ {1,..., µ} : x ≡ y ⇐⇒ σE(x) ≡ σE(y) ⇐⇒ σR(x) ≡ σR(y).
Remark 3.2.8. Suppose that (abc) is a 3-cycle of σR. Then if a ≡ b, we must
have σR(a) = b ≡ σR(b) = c, and similarly if a ≡ c or b ≡ c. Hence either
a,b,c are in the same block or they are all in different blocks.
We will now give an explicit example of how one checks for block systems.
Example 3.2.8. Consider the group Γ10,20 (see Table 3.1). There are two possibilities
for block systems. The block sizes are either 5+5 or 2+2+2+2+2.
We have σE = (12)(34)(56)(78)(910) and σR = (1)(235)(467)(8910). By
Remark 3.2.8 it is clear that we can not have a 5 + 5 block system. How about
2 + 2 + 2 + 2 + 2? Suppose that 1 ≡ x, then 1 = σR(1) ≡ σR(x), hence the
block containing 1 must contain x and σR(x). Since the block size is 2 we must
have σR(x) = x or σR(x) = 1, but the only fixed element of σR is 1, hence it is
impossible to have a 2 + 2 + 2 + 2 + 2 block system also.
We conclude that there is no supergroup between Γ10,20 and PSL(2,Z). And
in view of Proposition 3.2.1, Example 3.2.3 and Remark 3.2.5 we have shown
that there exist newforms on Γ10,20.
Remark 3.2.9. The groups we are primarily interested in are those which are
not congruence subgroups and that satisfy Prop. 3.2.1. Up to index 11 (which
is as far we have checked) this includes all groups G which satisfy (G ∗ ) = (G)
except for Γ10,8,3 (on which the symmetry α permutes the cusps). All these
groups have been investigated with respect to possible supergroups (note that
there can be no supergroup of groups with prime index).
Of the groups under consideration we found no supergroups for: all groups
of index 10, the groups Γ9,9;1, Γ9,20, Γ9,15 and Γ9,15 of index 9, Γ8,8;1 and Γ8,8;3
of index 8. On all these groups we claim that there exist newforms.
117

3.2.2 Congruence subgroups
Let G ⊆ PSL(2,Z) be a subgroup of index µ and generalized level N. It is
known that if G is a congruence subgroup then G has level N (cf. [104]).
We also remark that any PSL(2,Z)-conjugate of a congruence group is also a
congruence subgroup since Γ(N) is normal in PSL(2,Z).
Remark 3.2.10. Observe that if G has index µ and is a congruence subgroup
of level N, then µ must divide the index of Γ(N) in Γ(1) = PSL(2,Z):
[Γ(N) : Γ(1)] = 1
2 N3 −2
∏ 1 − p
p|N
, (3.3)
where the product is taken over all prime divisors of N (valid for N ≥ 3, for
N = 2 drop the factor 1
2 ). Hence if this is not the case, then G can not be a
congruence subgroup. The definitions of the groups that appear in the next two
examples are given in Table 3.1.
Example 3.2.9. Consider the groups Γ7,6, Γ7,10and Γ7,12 of type (7; 0,2,1,1)
and generalized levels 6, 10 and 12 respectively. From (3.3) we see that the
index of Γ(6) in PSL(2,Z) is 72, and the index of Γ(10) in PSL(2,Z) is 360.
Hence since 7 ∤ 72 and 7 ∤ 360, all the above groups are noncongruence subgroups.
Example 3.2.10. Consider the group Γ9,8 of type (9; 0,2,3,0) and generalized
level 8. The index of Γ(8) in Γ(1) is 192, and since 9 ∤ 192 the group Γ9,8 is a
noncongruence subgroup.
Clearly this fact can not tell us that a given group is a congruence subgroup,
but it is even the case that in general it can not even help us to determine
that a given group is not a congruence subgroup. E.g. the group Γ9,6;1 of
type (9; 0,2,3,0) and generalized level 6 is not a congruence subgroup even
though 9|72. But actually the group Γ9,6;2 with the same type is a congruence
subgroup. So the type and generalized level are not enough to decide if a
group is a congruence group or not.
Fortunately there is a stronger theorem that give us necessary and sufficient
conditions on the legitimate pair (σE,σR) in order for the corresponding subgroup
to be a congruence subgroup. The following theorem is reformulated
using our notation (and remember that we use an anti-homomorphism).
Theorem 3.4. (Hsu [47, thm. 3.1])
Suppose that (σE,σR) is a legitimate pair with corresponding group Γ and
that σS = σRσE has order N = 2 k m, where m is odd and k ≥ 0. Let a and b
be inverses of 2 and 5 mod N respectively (i.e. 2a ≡ 5b ≡ 1 mod N). We now
have three cases:
118

(a) N is odd, i.e. k = 0. Then Γ is a congruence subgroup if and only
if
1 = σRσSσRσ 1−a3
S .
(b) N is a power of 2, i.e. m = 1 : Let σS ˜ = σ 20
S (σRσS) b σ −5
S
σ −1
R .
Then Γ is a congruence subgroup if and only if the following three
relations hold:
σ −1
E ˜σSσE = ˜σ −1
S , (b:1)
˜σ −1
S (σRσS) ˜σS = (σRσS) 25 , (b:2)
˜σS (σRσS) 5 3 σE
= 1. (b:3)
(c) N has both an even and an odd part, i.e. if m ≥ 3 and k ≥ 1.
Let c be the unique integer mod N such that c ≡ 0 mod 2 k and
c ≡ 1 mod m, and let d be the unique integer mod N such that
d ≡ 0 mod m and d ≡ 1 mod 2 k . Let α = σ c S , β = (σRσS) c , γ = σ d,
S ,
ρ = (σRσS) d, , π = γ 20 ρ b γ −4 ρ −1 , ζ = αβ −1 α and ξ = γρ −1 γ.
Then Γ is a congruence subgroup if and only if the following 7
relations hold:
3.3 Numerical implementation
αρ = ρα, (c:1)
ζ 4 = 1, (c:2)
ζ 2 = β −1 α 3 , (c:3)
ζ 2 = β 2 α −a 3 , (c:4)
ξ 2 = πρ 5 ξ 3 , (c:5)
ξ −1 πξ = π −1 , (c:6)
π −1 ρπ = ρ 25, . (c:7)
Our goal is to investigate Maass waveforms on non-congruence subgroups,
and thus we must find examples of such groups.
Let N be a positive integer. By Theorem 3.2 we see that to find all subgroups
(up to conjugation) of index N in PSL(2,Z) we first need to find all equivalence
classes of legitimate pairs (σE,σR) in SN. To generate all elements of
SN is numerically speedy, and there exist a number of fast algorithms (cf.,
e.g. [49]).
The computationally hard part here is to identify the conjugates of legiti-
119

mate pairs, i.e. given a list of legitimate pairs, find out the ones which are conjugates
of each other. We know that conjugation preserves the cycle-structure,
i.e. if σ1 = σ −1 σ2σ then σ1 and σ2 have the same set of cycles. If σ conjugates
two pairs ( mod 1) we know first of all that σ(1) = 1, hence there are
only (N − 1)! choices, and we can narrow the number of possibilities down
further by studying the cycle structures of σE and σR. Unfortunately it seems
that without calculating the details by hand for a specific value of N the problem
is still hard even with as much automated simplifications as possible.
Once we have a list of non-conjugated legitimate pairs we use Theorem
3.4 to tell which pairs correspond to congruence subgroups. Then in order
to apply the usual Maass waveform algorithms (cf. Section 1.3) for the corresponding
subgroup of PSL(2,Z) we use Remark 3.2.1 to compute a set of
coset representatives.
Example 3.3.1. We will see how to explicitly compute representatives for the
PSL(2,Z)-conjugacy classes of the subgroups of type (7;0;1,1). First we observe
that σE consists of one fixed element and 3 2-cycles. The letter 1 is
special due to our convention in Theorem 3.2, but since we are only interested
in representatives we are free to conjugate the fixed element of σE to 1 (cf. Remark
3.2.2), hence, without loss of generality, we can assume that
σE = (1)(23)(45)(67).
Remember that the equivalence we consider (∼1) amounts to finding a permutation
π which preserves 1 and now that we have fixed σE, such π must
also preserve σE. Any choices we now make for σR are made modulo such a
permutation.
Now, we know that σR has one fixed element and 2 3-cycles. Since the
numbers 2,3,...,7 are now on equal footing we can assume that σR(2) = 2
and we can write
σR = (2)(1bc)(de f ),
and we have to choose the 3-cycles in turn. The elements 4,5,6,7 are equivalent,
while 3 is essentially different. There are thus 2 cases for b. Either b = 3
or b = 4. If b = 3, then we can assume c = 4 (and without loss of generality
d = 5) and we have
= (2)(134)(5e f ).
σR1
In case b = 4, then 6,7 are equivalent, and we observe that since (σE,σR) must
be transitive we can not have c = 5. Hence either c = 3, or c = 6 (and, wlog
d = 5 in both cases). We have
120
σR2
σR3
= (2)(143)(5e f ), and
= (2)(146)(3e f ).

We now have to choose the last 3-cycle. For the last cycle of σR1 and σR2 ,
clearly 6 and 7 are equivalent, and wlog e = 6 and f = 7 and we get
σR1
σR2
= (2)(134)(567), and
= (2)(143)(567).
For σR3 = (2)(146)(5e f ), on the other hand, there are two possible (inequivalent)
choices e = 5, f = 7 or e = 7, f = 5, corresponding to
σR3
σR4
= (2)(146)(357), and
= (2)(146)(375).
We thus have 4 inequivalent legitimate pairs: σE,σR j
subgroups G j, j = 1,2,3,4. The corresponding σS j = σR j σE are
σS1
σS2
σS3
σS4
= (132465)(7),
= (146532)(7),
= (147)(2563),
= (14327)(56).
, corresponding to 4
And we see that the generalized levels are 6, 6, 12 and 10. Note that σ ∗ R1 =
σEσ 2 R1 σE = π −1 σR2 π ∼1 σR2 where π = (32)(45). Hence G∗ 1 = G2. The corresponding
groups in Table 3.1 are Γ7,6, Γ ∗ 7,6 , Γ7,12 and Γ7,10. Fundamental
domains of these groups, with the combinatorial structure indicated, can be
seen in Figure 3.1.
3.4 Numerical Results
3.4.1 Subgroups
In this section we will give examples of results obtained. Table 3.1 contains
lists of specifications of conjugacy classes of subgroups of PSL(2,Z) of index
less than or equal to 12. For each conjugacy class we have chosen a representative
group, Γµ,l;i, for which we have also given the corresponding permutations.
Here µ is the index, l is the generalized level and i is an arbitrary (but
fixed by the program) enumeration of the conjugacy classes.
To show which groups have a reflectional symmetry (cf. Prop. 3.2.1) we
also list the equivalence class of G ∗ = JGJ. In this column we use the notation
(G ∗ ) to say that (G ∗ ) = (G), and (G) to say that (G ∗ ) = (G) for the given
group G.
121

Figure 3.1: Fundamental domains for groups of type (7;0,1,1)
122
1 3 2 4 6 5
(a) Γ7,6
1 4 6 5 3 2
1 4 7
6
3
5
2
(c) Γ7,12
7
(b) Γ ∗ 7,6
1 4 3 2 6
7
5
(d) Γ7,10
7

3.4.2 Newforms
We have checked for Maass waveforms (newforms) on all the listed groups
with index ≤ 11. As expected we found newforms on all congruence subgroups
and all cycloidal subgroups. 1
We also found newforms on all noncongruence, noncycloidal subgroups
which satisfy Proposition 3.2.1. In this case, the number of newforms that we
found is approximately one half of what is indicated by Weyl’s law for the
surface G\H, as is to be expected for the α-odd space L 2 − (G\H). (In this
vein, cf. [88, pp. 448 (footnote 1), 670 (middle)].) Since our investigation has
been directed at investigating as many groups as possible, we have generally
not taken R beyond 10 or so for each group. (That the Weyl ratios are only
approximately 50% is thus not very disturbing; taking R larger undoubtedly
produces values much closer to 50%.)
In the case of noncongruence, noncycloidal groups which do not satisfy
Proposition 3.2.1, we found no newforms whatsoever. An example of a group
satisfying (G ∗ ) = (G), but that violates the cusp class preserving condition
is Γ10,8;3. This indicates that the Phillips-Sarnak conjecture is true even for
subgroups of the modular group.
When testing for existence/nonexistence of newforms we ran the program
up to height R = 10 for indexes ≤ 10 and up to R = 5 for index 11.
In all cases of non-congruence, non-cycloidal groups that we have checked,
all newforms found have been of α−odd type. For cycloidal groups, we found
both α-even and α-odd newforms, exactly as one would expect from the trace
formula. Cf. footnote 1.
Tables 3.2-3.6 contain examples of eigenvalues for various noncongruence
subgroups.
Further data than that which is presented in this section is available from
the author upon request.
3.4.3 Fourier coefficients
We have also investigated the distribution of Fourier coefficients in a few examples.
Figure 3.3 (b) gives a histogram of 12422 Fourier coefficients (they
are real and we normalized them to have mean 0 and standard deviation 1) of a
newform with eigenvalue R = 3.34168098977 on Γ7,12. In the same figure we
also plotted the Gaussian curve with the same mean and standard deviation.
We have also studied the Fourier coefficients of a newform on Γ10,8;1 with
eigenvalue R = 2.40931877484576. Here one should notice that the symmetry
J acts as a shifted reflection on Γ10,8;1 (i.e. Γ ∗ 10,8;1 ∼ S−5 Γ10,8;1S 5 ) and hence
the Fourier coefficients will not be real in this case, but have arguments shifted
1 The Eisenstein series and scattering determinant for cycloidal groups are readily seen to be
essentially the same as for PSL(2,Z). Cf. [105] and see also [100].
123

Figure 3.2: Plot of Fourier coefficients in the complex plane.
2
1.5
1
0.5
0
-0.5
-1
-1.5
”14412 Coefficients for R = 2.0493 . . . on Γ10,8;1”
-2
-3 -2 -1 0 1 2 3 4
by multiples π
8 . The surprising thing here is that it seems that the different
residue classes modulo 8 also have different standard deviation, as can be
seen in figure 3.2, which shows a plot in the complex plane of the first 14412
Fourier coefficients. The coefficients were shifted back to the real line and
then normalized with appropriate standard deviation (according to the residue
class n mod 8). Figure 3.3(a) contains a histogram and a Gaussian fit to the
resulting sequence.
In both of these cases the fit to the Gaussian looks good, but more data
has to be analyzed before one can draw any conclusions. This is certainly
something that needs further investigation.
124

Figure 3.3: Distribution of Fourier Coefficients
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
14412 Coefficients
Γ 10,8;1
R=2.0493
0
−4 −3 −2 −1 0 1 2 3 4 5
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
(a) At the group Γ10,8;1
12422 Coefficients
Γ 7,12
R=3.341...
0
−4 −3 −2 −1 0 1 2 3 4 5
(b) At the group Γ7,12
125

126
Table 3.1: Conjugacy classes of subgroups of PSL(2,Z) with index ≤ 12
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
Index=1 (1 equivalence class)
(Γ2,2) 1 0 2 0 (G) T (1 2) ( 1)( 2) ( 1 2)
Index=3 (2 equivalence classes)
(Γ3,3) 1 3 0 0 (G) T (1)(2)(3) (1 2 3) (1 2 3)
(Γ3,2) 2 1 0 0 (G) T (1)(2 3) (1 2 3) (1 2)(3)
Index=4 (2 equivalence classes)
(Γ4,4) 1 2 1 0 (G) T (1)(2)(3 4) (1 2 4)(3) (1 2 4 3)
(Γ4,3) 2 0 1 0 (G) T (1 2)(3 4) (1)(2 3 4) (1 3 2)(4)
Index=5 (1 equivalence class)
(Γ5,5) 1 1 2 0 (G) T (1)(2 3)(4 5) (1 3 5)(2)(4) (1 3 2 5 4)
Index=6 (10 equivalence classes)
(Γ6,6;1) 1 0 0 1 (G) T (1 2)(3 4)(5 6) (1 3 5)(2 4 6) (1 4 5 2 3 6)
(Γ6,6;2) 1 0 3 0 (G) T (1 2)(3 4)(5 6) (1)(2 4 6)(3)(5) (1 4 3 6 5 2)
(Γ6,6;3) 1 4 0 0 (G) T (1)(2)(3)(4)(5 6) (1 2 5)(3 4 6) (1 2 5 3 4 6)
(Γ6,5) 2 2 0 0 (G) T (1)(2)(3 4)(5 6) (1 2 3)(4 5 6) (1 2 3 5 4)(6)
(Γ6,3) 2 2 0 0 (G) T (1)(2)(3 4)(5 6) (1 3 5)(2 4 6) (1 3 6)(2 4 5)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

127
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ6,4;1) 2 2 0 0 (G) T (1)(2)(3 4)(5 6) (1 3 6)(2 5 4) (1 3 2 5)(4 6)
(Γ6,4;2) 3 0 0 0 (G) T (1 2)(3 4)(5 6) (1 2 3)(4 5 6) (1 3 5 4)(2)(6)
(Γ6,4;3) 3 0 0 0 (G) T (1 2)(3 4)(5 6) (1 3 2)(4 6 5) (1)( 2 3 6 4)(5)
(Γ6,4;4) 3 0 0 0 (G) T (1 2)(3 4)(5 6) (1 4 3)(2 6 5) (1 6 2 4)(3)(5)
(Γ6,2) 3 0 0 0 (G) T (1 2)(3 4)(5 6) (1 3 6)(2 5 4) (1 5)(2 3)(4 6)
Index=7 (6 equivalence classes)
(Γ7,7) 1 3 1 0 (G ∗ ) T (1)(2)(3)(4 5)(6 7) (1 2 6)(3 5 7)(4) (1 2 6 3 5 4 7)
(Γ7,6) 2 1 1 0 (G ∗ ) F (1)(2 3)(4 5)(6 7) (1 3 4)(2)(5 6 7) (1 3 2 4 6 5)(7)
(Γ7,12) 2 1 1 0 (G) F (1)(2 3)(4 5)(6 7) (1 4 6)(2)(3 5 7) (1 4 7)(2 5 6 3)
(Γ7,10) 2 1 1 0 (G) F (1)(2 3)(4 5)(6 7) (1 4 7)(2)(3 6 5) (1 4 3 2 6)(5 7)
Index=8 (6 equivalence classes)
(Γ8,8;1) 1 2 2 0 (G) F (1)(2)(3 4)(5 6)(7 8) (1 2 7)(3)(4 6 8)(5) (1 2 7 4 3 6 5 8)
(Γ8,8;2) 1 2 2 0 (G ∗ ) T (1)(2)(3 4)(5 6)(7 8) (1 4 7)(2 6 8)(3)(5) (1 4 3 7 2 6 5 8)
(Γ8,8;3) 1 2 2 0 (G) F (1)(2)(3 4)(5 6)(7 8) (1 4 8)(2 7 6)(3)(5) (1 4 3 8 6 5 2 7)
(Γ8,7) 2 0 2 0 (G) T (1 2)(3 4)(5 6)(7 8) (1)(2 4 5)(3)(6 7 8) (1 4 3 5 7 6 2)(8)
(Γ8,4) 2 0 2 0 (G) T (1 2)(3 4)(5 6)(7 8) (1)(2 5 7)(3)(4 6 8) (1 5 8 2)(3 6 7 4)
(Γ8,6) 2 0 2 0 (G) T (1 2)(3 4)(5 6)(7 8) (1)(2 5 8)(3)(4 7 6) (1 5 4 3 7 2)(6 8)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

128
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
Index=9 (14 equivalence classes)
(Γ9,9;1) 1 1 0 1 (G) F (1)(2 3)(4 5)(6 7)(8 9) (1 2 4)(3 6 8)(5 7 9) (1 2 6 9 3 4 7 8 5)
(Γ9,9;2) 1 1 3 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9) (1 3 8)(2)(4)(5 7 9)(6) (1 3 2 8 5 4 76 9)
(Γ9,9;3) 1 5 0 0 (G) T (1)(2)(3)(4)(5)(6 7)(8 9) (1 2 6)(3 4 8)(5 7 9) (1 2 6 9 3 4 8 5 7)
(Γ9,8) 2 3 0 0 (G ∗ ) F (1)(2)(3)(4 5)(6 7)(8 9) (1 2 4)(3 5 6)(7 8 9) (1 2 4 6 8 7 3 5)(9)
(Γ9,6;1) 2 3 0 0 (G) F (1)(2)(3)(4 5)(6 7)(8 9) (1 2 4)(3 6 8)(5 7 9) (1 2 4 7 8 5)(36 9)
(Γ9,14) 2 3 0 0 (G) F (1)(2)(3)(4 5)(6 7)(8 9) (1 2 4)(3 6 8)(5 9 7) (1 2 4 9 3 6 5)(7 8)
(Γ9,20) 2 3 0 0 (G) F (1)(2)(3)(4 5)(6 7)(8 9) (1 4 6)(2 5 8)(3 7 9) (1 4 8 3 7)(2 56 9)
(Γ9,6;2) 2 3 0 0 (G) T (1)(2)(3)(4 5)(6 7)(8 9) (1 4 6)(2 7 9)(3 8 5) (1 4 3 8 2 7)(5 6 9)
(Γ9,15) 3 1 0 0 (G) F (1)(2 3)(4 5)(6 7)(8 9) (1 2 4)(3 5 6)(7 8 9) (1 2 5)(3 4 6 8 7)(9)
(Γ9,6;3) 3 1 0 0 (G) F (1)(2 3)(4 5)(6 7)(8 9) (1 2 4)(3 6 5)(7 8 9) (1 2 6 8 7 5)(3 4)(9)
(Γ9,7) 3 1 0 0 (G) F (1)(2 3)(4 5)(6 7)(8 9) (1 2 4)(3 6 7)(5 8 9) (1 2 6 3 4 8 5)(7)(9)
(Γ9,12) 3 1 0 0 (G) F (1)(2 3)(4 5)(6 7)(8 9) (1 2 4)(3 6 8)(5 9 7) (1 2 6 5)(3 4 9)(7 8)
Index=10 (26 equivalence classes)
(Γ10,10;1) 1 0 1 1 (G) F (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 3 5)(4 7 9)(6 8 10) (1 3 7 10 4 5 8 9 6 2)
(Γ10,10;2) 1 0 4 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 4 9)(3)(5)(6 8 10)(7) (1 4 3 9 6 5 8 7 10 2)
(Γ10,10;3) 1 4 1 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10) (1 2 7)(3 4 9)(5)(6 8 10) (1 2 7 10 3 4 9 6 5 8)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

129
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ10,10;4) 1 4 1 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10) (1 2 7)(3 6 9)(4 8 10)(5) (1 2 7 10 3 6 5 9 4 8)
(Γ10,10;5) 1 4 1 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10) (1 2 7)(3 6 10)(4 9 8)(5) (1 2 7 4 9 3 6 5 10 8)
(Γ10,9;1) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 2 5)(3)(4 6 7)(8 9 10) (1 2 5 7 9 8 4 3 6)(10)
(Γ10,9;2) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 4 7)(2 5 8)(3)(6 9 10) (1 4 3 7 2 5 9 6 8)(10)
(Γ10,9;3) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 4 5)(2 6 7)(3)(8 9 10) (1 4 3 5 7 9 8 2 6)(10)
(Γ10,12;1) 2 2 1 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 2 5)(3)(4 7 9)(6 8 10) (1 2 5 8 9 6)(3 7 10 4)
(Γ10,12;2) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 5 7)(2 6 9)(3)(48 10) (1 5 9 4 3 8)(2 6 7 10)
(Γ10,8;1) 2 2 1 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 2 5)(3)(4 7 10)(6 9 8) (1 2 5 9 4 3 7 6)(8 10)
(Γ10,8;2) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 4 7)(2 5 9)(3)(6 8 10) (1 4 3 7 10 2 5 8)(6 9)
(Γ10,21;1) 2 2 1 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 4 7)(2 5 9)(3)(6 10 8) (1 4 3 7 6 9 8)(2 5 10)
(Γ10,21;2) 2 2 1 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10) (1 5 7)(2 8 10)(3)(49 6) (1 5 4 3 9 2 8)(6 7 10)
(Γ10,5) 2 2 1 0 (G) T (1)(2)(3 4)(5 6)(7 8)(9 10) (1 5 7)(2 8 10)(3)(4 6 9) (1 5 9 2 8)(3 6 7 10 4)
(Γ10,20) 3 0 1 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 3 5)(4 6 7)(8 9 10) (1 3 6 2)(4 5 7 9 8)(10)
(Γ10,14) 3 0 1 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 3 5)(4 7 6)(8 9 10) (1 3 7 9 8 6 2)(4 5)(10)
(Γ10,8;3) 3 0 1 0 (G) a F (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 3 5)(4 7 8)(6 9 10) (1 3 7 4 5 9 6 2)(8)(10)
(Γ10,30) 3 0 1 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10) (1)(2 3 5)(4 7 10)(6 9 8) (1 3 7 6 2)(4 5 9)(8 10)
Index=11 (26 equivalence classes)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

130
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ11,11;1) 1 3 2 0 (G) F (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 8 10)(2 9 7)(3 5 11)(4)(6) (1 8 7 6 2 9 10 3 5 4 11)
(Γ11,11;2) 1 3 2 0 (G) F (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 5 10)(2 11 8)(3 9 7)(4)(6) (1 5 4 10 8 7 6 3 9 2 11)
(Γ11,11;3) 1 3 2 0 (G ∗ ) F (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 5 10)(2 9 3)(4)(6)(7 11 8) (1 5 4 10 8 3 2 9 7 6 11)
(Γ11,11;4) 1 3 2 0 (G ∗ ) T (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 2 10)(3 5 9)(4)(6)(7 11 8) (1 2 10 8 3 5 4 9 7 6 11)
(Γ11,11;5) 1 3 2 0 (G ∗ ) F (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 2 8)(3 9 11)(4)(5 10 7)(6) (1 2 8 11 7 6 5 4 10 3 9)
(Γ11,11;6) 1 3 2 0 (G ∗ ) F (1)(2)(3)(4 5)(6 7)(8 9)(10 11) (1 8 10)(2 11 7)(3 9 5)(4)(6) (1 8 5 4 3 9 10 7 6 2 11)
(Γ11,28;1) 2 1 2 0 (G) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 8 6)(2)(3 9 11)(4)(5 10 7) (1 8 11 7)(2 9 6 5 4 10 3)
(Γ11,28;2) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 6 3)(2)(4)(5 10 8)(7 11 9) (1 6 11 8 7 3 2)(4 10 9 5)
(Γ11,10;1) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 3 6)(2)(4)(5 7 11)(8 10 9) (1 3 2 6 11 9 10 5 4 7)(8)
(Γ11,10;2) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 6 3)(2)(4)(5 7 11)(8 10 9) (1 6 11 9 10 5 4 7 3 2)(8)
(Γ11,10;3) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 6 8)(2)(3 7 5)(4)(9 11 10) (1 6 5 4 3 2 7 8 11 9)(10)
(Γ11,18;1) 2 1 2 0 (G) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 8 6)(2)(3 5 11)(4)(7 9 10) (1 8 10 3 2 5 4 11 7)(6 9)
(Γ11,18;2) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 3 6)(2)(4)(5 9 11)(7 10 8) (1 3 2 6 10 5 4 9 7)(8 11)
(Γ11,30) 2 1 2 0 (G ∗ ) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 6 8)(2)(3 9 10)(4)(5 7 11) (1 6 11 3 2 9)(4 7 8 10 5)
(Γ11,24;1) 2 1 2 0 (G) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 8 6)(2)(3 5 10)(4)(7 11 9) (1 8 7)(2 5 4 10 9 6 11 3)
(Γ11,24;2) 2 1 2 0 (G) F (1)(2 3)(4 5)(6 7)(8 9)(10 11) (1 6 8)(2)(3 9 11)(4)(5 10 7) (1 6 5 4 10 3 2 9)(7 8 11)
Index=12 (81 equivalence classes)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

131
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ12,12;1) 1 2 0 1 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 3)(4 7 10)(5 8 11)(6 9 12) (1 2 3 7 11 6 8 10 12 5 9 4)
(Γ12,12;2) 1 2 0 1 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 7)(4 11 8)(6 12 9) (1 3 11 9 7 4 5 12 8 2 10 6)
(Γ12,12;3) 1 2 0 1 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 7)(4 12 9)(6 11 8) (1 3 12 8 2 10 4 5 11 9 7 6)
(Γ12,12;4) 1 2 0 1 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 9 6)(4 12 8)(7 10 11) (1 3 12 7 4 5 2 9 11 8 10 6)
(Γ12,12;6) 1 2 0 1 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 9 4)(6 12 8)(7 10 11) (1 3 2 9 11 8 10 4 5 12 7 6)
(Γ12,12;7) 1 2 3 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 9 11)(2 10 8)(3)(4 6 12)(5)(7) (1 9 8 7 2 10 11 4 3 6 5 12)
(Γ12,12;9) 1 2 3 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 9 11)(2 12 8)(3)(4 6 10)(5)(7) (1 9 4 3 6 5 10 11 8 7 2 12)
(Γ12,12;11) 1 2 3 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 11)(3)(4 6 10)(5)(7)(8 12 9) (1 2 11 9 4 3 6 5 10 8 7 12)
(Γ12,12;13) 1 2 3 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 4 11)(2 6 10)(3)(5)(7)(8 12 9) (1 4 3 11 9 2 6 5 10 8 7 12)
(Γ12,12;15) 1 2 3 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 4 11)(2 10 6)(3)(5)(7)(8 12 9) (1 4 3 11 9 6 5 2 10 8 7 12)
(Γ12,12;16) 1 2 3 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 4 11)(2 9 8)(3)(5)(6 10 12)(7) (1 4 3 11 6 5 10 8 7 2 9 12)
(Γ12,12;17) 1 6 0 0 (G) F (1)(2)(3)(4)(5)(6)(7 8)(9 10)(11 12) (1 2 7)(3 12 6)(4 8 10)(5 9 11) (1 2 7 10 11 6 3 12 5 9 4 8)
(Γ12,12;18) 1 6 0 0 (G ∗ ) F (1)(2)(3)(4)(5)(6)(7 8)(9 10)(11 12) (1 2 7)(3 811)(4 9 12)(5 10 6) (1 2 7 11 4 9 6 5 10 12 3 8)
(Γ12,12;20) 1 6 0 0 (G) T (1)(2)(3)(4)(5)(6)(7 8)(9 10)(11 12) (1 2 7)(3 104)(5 12 6)(8 11 9) (1 2 7 11 6 5 12 9 4 3 10 8)
(Γ12,8;1) 2 0 0 1 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 7)(4 12 9)(6 11 8) (1 10 4 5 11 9 7 6)(2 3 12 8)
(Γ12,6;1) 2 0 0 1 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 9)(4 6 7)(8 12 10) (1 11 10 2 3 6)(4 5 7 12 9 8)
(Γ12,10;1) 2 0 0 1 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 12)(4 6 7)(8 11 9) (1 10 8 4 5 7 11 2 3 6)(9 12)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

132
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ12,11;1) 2 0 0 1 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 9 6)(4 10 7)(8 11 12) (1 9 7 11 8 4 5 2 3 10 6)(12)
(Γ12,9;1) 2 0 0 1 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 8)(4 11 9)(6 7 12) (1 10 4 5 7 2 3 11 6)(8 12 9)
(Γ12,35;1) 2 0 3 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 10 7)(3)(4 11 8)(5)(6 12 9) (1 10 6 5 12 8 2)(3 11 9 7 4)
(Γ12,11;2) 2 0 3 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 9 6)(3)(4 10 7)(5)(8 11 12) (1 9 7 11 8 4 3 10 6 5 2)(12)
(Γ12,11;3) 2 0 3 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 6 9)(3)(4 7 10)(5)(8 12 11) (1 6 5 9 4 3 7 12 8 10 2)(11)
(Γ12,10;2) 2 0 3 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 10 4)(3)(5)(6 11 8)(7 12 9) (1 10 7 6 5 11 9 4 3 2)(8 12)
(Γ12,8;2) 2 0 3 0 (G) F (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 9 6)(3)(4 12 8)(5)(7 10 11) (1 9 11 8 10 6 5 2)(3 12 7 4)
(Γ12,9;2) 2 0 3 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1)(2 8 10)(3)(4 9 11)(5)(6 12 7) (1 8 6 5 12 4 3 9 2)(7 10 11)
(Γ12,11;4) 2 4 0 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 6 8)(4 7 12)(9 11 10) (1 2 5 8 12 10 11 4 7 3 6)(9)
(Γ12,9;3) 2 4 0 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 11 9)(3 4 6)(8 12 10) (1 5 3 4 6 7 12 9 8)(2 11 10)
(Γ12,8;3) 2 4 0 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 10 7)(4 11 8)(6 12 9) (1 2 5 12 8 3 10 6)(4 11 9 7)
(Γ12,35;2) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 12 9)(4 10 7)(6 11 8) (1 2 5 11 9 7 6)(3 12 8 4 10)
(Γ12,10;3) 2 4 0 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 10 12)(3 4 6)(8 11 9) (1 5 3 4 6 7 11 2 10 8)(9 12)
(Γ12,11;6) 2 4 0 0 (G ∗ ) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 10 8)(3 4 6)(9 11 12) (1 5 3 4 6 7 2 10 11 9 8)(12)
(Γ12,10;5) 2 4 0 0 (G) T (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 10 4)(6 11 8)(7 12 9) (1 2 5 11 9 4 3 10 7 6)(8 12)
(Γ12,35;3) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 10 8)(3 6 12)(4 11 9) (1 5 12 9 8)(2 10 4 11 3 6 7)
(Γ12,6;2) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 11 4)(6 10 8)(7 12 9) (1 2 5 10 7 6)(3 11 9 8 12 4)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

133
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ12,6;3) 2 4 0 0 (G) T (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 11 8)(3 12 9)(4 6 10) (1 5 10 3 12 8)(2 11 9 4 6 7)
(Γ12,11;8) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 4 7)(6 8 12)(9 11 10) (1 2 5 8 3 4 7 12 10 11 6)(9)
(Γ12,9;5) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 2 5)(3 10 8)(4 11 9)(6 7 12) (1 2 5 7 3 10 4 11 6)(8 12 9)
(Γ12,6;4) 2 4 0 0 (G) F (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 12 6)(3 10 8)(4 11 9) (1 5 2 12 9 8)(3 10 4 11 6 7)
(Γ12,8;5) 2 4 0 0 (G) T (1)(2)(3)(4)(5 6)(7 8)(9 10)(11 12) (1 5 7)(2 12 6)(3 8 10)(4 9 11) (1 5 2 12 4 9 3 8)(6 7 10 11)
(Γ12,18;1) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 3)(4 9 6)(5 10 7)(8 11 12) (1 2 3 9 7 11 8 5 4)(6 10)(12)
(Γ12,30;1) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 3)(4 7 9)(5 8 10)(6 12 11) (1 2 3 7 10 4)(5 12 6 8 9)(11)
(Γ12,42;1) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 3)(4 7 11)(5 8 10)(6 9 12) (1 2 3 7 10 12 4)(5 9)(6 8 11)
(Γ12,6;5) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 9)(4 6 7)(8 12 10) (1 3 6)(2 11 10)(4 5 7 12 9 8)
(Γ12,12;21) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 4)(6 11 8)(7 12 9) (1 3 2 10 7 6)(4 5 11 9)(8 12)
(Γ12,42;2) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 12)(4 6 7)(8 11 9) (1 3 6)(2 10 8 4 5 7 11)(9 12)
(Γ12,24;1) 3 2 0 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 10 8)(4 6 7)(9 11 12) (1 3 6)(2 10 11 9 8 4 5 7)(12)
(Γ12,28;1) 3 2 0 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 9 6)(4 10 7)(8 11 12) (1 3 10 6)(2 9 7 11 8 4 5)(12)
(Γ12,4;1) 3 2 0 0 (G) T (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 8)(4 10 7)(6 12 9) (1 3 10 6)(2 11 9 7)(4 5 12 8)
(Γ12,24;3) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 9 4)(6 10 7)(8 11 12) (1 3 2 9 7 11 8 6)(4 5 10)(12)
(Γ12,10;6) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 2 3)(4 8 10)(5 9 6)(7 12 11) (1 2 3 8 12 7 10 6 9 4)(5)(11)
(Γ12,60;1) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 8)(4 12 9)(6 10 7) (1 3 12 8 6)(2 11 9 7)(4 5 10)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

134
Γµ,l,i h e2 e3 g G ∗ C a σE σR σS
(Γ12,10;7) 3 2 0 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 4 10)(6 11 8)(7 12 9) (1 3 10 7 6)(2 4 5 11 9)(8 12)
(Γ12,30;2) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 6 8)(4 7 12)(9 11 10) (1 3 7 2 6)(4 5 8 12 10 11)(9)
(Γ12,10;9) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 6 11)(4 7 8)(9 12 10) (1 3 7 4 5 11 10 12 2 6)(8)(9)
(Γ12,10;10) 3 2 0 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 12 6)(4 7 8)(9 11 10) (1 3 7 4 5 2 12 10 11 6)(8)(9)
(Γ12,6;6) 3 2 0 0 (G) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 9)(4 10 8)(6 7 12) (1 3 10 2 11 6)(4 5 7)(8 12 9)
(Γ12,18;2) 3 2 0 0 (G ∗ ) F (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 9)(4 12 6)(7 10 8) (1 3 12 9 8 10 2 11 6)(4 5)(7)
(Γ12,8;6) 3 2 0 0 (G) T (1)(2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 10)(4 8 6)(7 9 12) (1 3 8 9 2 11 7 6)(4 5)(10 12)
(Γ12,6;7) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 4)(2 7 12)(5 8 10)(6 9 11) (1 7 10 11 2 3)(4)(5 9)(6 8 12)
(Γ12,8;7) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 4)(2 9 6)(5 10 7)(8 11 12) (1 9 7 11 8 5 2 3)(4)(6 10)(12)
(Γ12,9;6) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 4)(2 8 5)(6 9 10)(7 12 11) (1 8 12 7 5 9 6 2 3)(4)(10)(11)
(Γ12,5;1) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 12 6)(4 7 8)(9 11 10) (1 12 10 11 6)(2 3 7 4 5)(8)(9)
(Γ12,3;1) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 9)(4 10 8)(67 12) (1 11 6)(2 3 10)(4 5 7)(8 12 9)
(Γ12,6;8) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 5 3)(2 9 11)(4 6 12)(7 8 10) (1 9 7 10 11 4)(2 5 12)(3 6)(8)
(Γ12,4;2) 4 0 0 0 (G) T (1 2)(3 4)(5 6)(7 8)(9 10)(11 12) (1 3 5)(2 11 10)(4 8 6)(79 12) (1 11 7 6)(2 3 8 9)(4 5)(10 12)
a This column shows if the listed group is a congruence subgroup (T) or not (F).

Table 3.2: Eigenvalues on Γ7,10 (2 cusps)
R |c(−1) + 1| a h1
2.98380890177954200 7E-12 2E-15
4.25864509747714400 1E-12 4E-13
4.89540660025545100 2E-13 4E-13
5.28884863158980700 1E-12 6E-14
6.15234838528998200 0E+00 2E-14
6.27199072998424200 2E-13 4E-13
6.76036517325738900 2E-13 2E-13
7.20603858428835700 4E-13 1E-14
7.55764967370570000 4E-13 7E-13
7.77456459246736200 6E-14 3E-14
8.50359685011489000 8E-13 8E-14
8.77366479019266900 3E-13 2E-12
8.88752983993304300 2E-13 2E-14
9.29204073179663000 5E-14 6E-13
9.53369526135354484 * 2E-14 3E-14
9.67736151045037300 4E-13 6E-14
10.04237676924058000 1E-10 2E-08
All newforms listed are odd with respect to the symmetry α : z ↦→ −¯z + 5.
* An oldform from PSL(2,Z).
a See Remark 3.2.6 for an explanation of the term 1.
135

Table 3.3: Eigenvalues on Γ9,20;1 (2 cusps)
R |c(−1) − e 2πi
10 | a h1
2.05495976063979 6E-11 1E-15
3.25563226951552 1E-12 2E-14
4.01743421448455 8E-13 1E-12
4.39990456866076 3E-13 9E-12
4.92041181302557 1E-13 5E-14
5.22354223285616 1E-11 6E-11
5.70296517107618 2E-14 3E-13
6.20783400944797 3E-12 6E-12
6.52618304206566 8E-13 2E-12
6.78783029888573 4E-13 5E-14
7.07791461588707 2E-12 1E-14
7.38877048107201 5E-15 8E-13
8.02743455884534 3E-10 3E-09
8.70656893384487 2E-13 7E-10
8.81164911527479 6E-15 9E-13
9.03359858422541 9E-11 3E-10
9.50035839336283 2E-11 4E-11
9.53369526135354 * 5E-09 7E-14
9.66113248208174 3E-11 3E-12
All newforms listed are odd with respect to the symmetry α : z ↦→ −¯z + 4.
* An oldform from PSL(2,Z).
a See Remark 3.2.6 for an explanation of the term e 2πi
10 .
136

Table 3.4: Eigenvalues of newforms on Γ9,15 (3 cusps)
R |c(−1) + 1| a h1
3.11274526127958 8E-13 2E-15
3.79758314524666 8E-14 1E-14
4.68277129730610 2E-14 3E-11
5.61420134556306 9E-12 3E-11
6.17840605444398 4E-14 1E-10
6.28023134316972 2E-11 4E-10
6.50085225008714 8E-10 2E-10
6.87257413990397 8E-09 6E-10
7.21783938535166 2E-10 9E-10
7.41980107735753 2E-11 4E-11
7.73811611339896 2E-12 4E-12
7.89543101394190 3E-10 1E-10
8.33429557635011 5E-11 6E-09
8.82147607539715 4E-08 2E-10
8.83035851126317 6E-09 2E-08
9.17711548223084 3E-10 7E-10
9.23041542661088 6E-08 3E-09
9.53369526135299 * 4E-09 5E-11
9.73675250028969 3E-09 3E-10
All newforms listed are odd with respect to the symmetry α : z ↦→ −¯z + 3.
* An oldform from PSL(2,Z).
a See Remark 3.2.6 for an explanation of the term 1.
137

Table 3.5: Eigenvalues of newforms on a cycloidal group of index 10
Γ10,10,1
R |c(−1) − e 9πi
5 | a |c(−1) − e 4πi
5 | a h1
1.56460530796110 e 6E-11 3E-15
2.11939343023923 o 1E-11 1E-12
2.50414532509967 e 1E-09 2E-09
2.91592762476298 e 6E-10 3E-14
3.00023468927910 o 2E-11 4E-15
3.10845111145684 o 1E-13 1E-11
3.56237482722476 e 6E-11 2E-15
3.65766022871834 o 1E-12 2E-13
3.88274850431287 o 2E-12 5E-11
4.10694094068138 e 6E-11 2E-14
4.55947270613401 o 4E-12 4E-14
4.74782424736486 o 3E-13 4E-15
4.76752492536872 e 6E-11 2E-14
4.84953817531720 e 1E-10 2E-12
e Even with respect to the symmetry α : z ↦→ −¯z + 9.
o Odd with respect to the symmetry α : z ↦→ −¯z + 9.
a See Remark 3.2.6 for an explanation of the terms e 9πi
5 and e 4πi
5 .
138

Table 3.6: Eigenvalues of newforms on groups of index 10
Γ10,12,1 R |c(−1) − e 2πi
3 | a h1
2.78187318285282 7E-12 5E-16
3.37184320841770 5E-11 7E-12
3.76035106954282 3E-11 8E-12
4.22985717336497 5E-13 6E-15
4.85352368236249 3E-13 1E-12
Γ10,8,1 R |c(−1) − e 3πi
4 | a h1
2.40931877484576 2E-10 3E-14
3.29803900236809 5E-09 2E-10
3.70953186018084 1E-11 3E-13
4.27459659497848 2E-10 6E-11
4.58646680301038 1E-12 2E-11
Γ10,21,2 R |c(−1) − e 3πi
7 | a h1
1.90995590589148 5E-09 2E-14
3.20456049497589 1E-08 6E-13
3.50475869850690 5E-09 4E-12
4.20753581320237 5E-09 6E-11
4.77744371781533 5E-08 8E-10
5.00937515698108 5E-09 3E-11
Γ10,20 R |c(−1) − i| a h1
3.08773394089329578 2E-10 1E-13
3.50709021769279694 7E-10 2E-14
4.34565944211539890 7E-10 3E-13
Γ10,14 R |c(−1) − e 5πi
7 | a h1
2.57440388633114603 0E+00 2E-15
3.79413397458154922 4E-11 2E-14
4.06173226279882993 4E-11 2E-11
4.68613704425140210 3E-11 1E-14
Γ10,30 R |c(−1) − e 3πi
5 | a h1
2.68171983329039 2E-10 4E-16
3.56451256590188 1E-10 4E-15
4.44205124980089 7E-10 1E-12
4.70189370885265 2E-10 1E-14
4.97437106689348 2E-10 2E-15
All newforms listed above are odd with respect to the action of J on the groups.
a See Remark 3.2.6 for an explanation of the different terms e ·πi
· .
139

4 An Algorithm for Whittaker’s W-function
4.1 Introduction
We present a new algorithm for computing the Whittaker function Wl,iR(x)
based on an integral representation similar to the integral representation of the
K-Bessel function,
KiR(x) = 1
∞
e
2 −∞
−xcosht · 1 · e iRt dt.
In our case, the factor “1” in front of the exponential, e iRt , in the integrand is
replaced with a confluent hypergeometric function, Ψ, with real parameters
and complex argument. The advantage with this approach is that we avoid the
computationally harder problem of evaluating the confluent hypergeometric
functions with complex parameters. We then turn the path of the integral to
a path of “almost stationary” phase, and the resulting integral turns out to be
well-suited for standard quadrature techniques.
It is clear that for this algorithm to be of any practical use we need efficient
algorithms for the confluent hypergeometric function with real parameters.
Fortunately there is an abundance of literature on this subject, and we will
present three different methods.
4.2 Presentation of the Algorithm
We use the integral formula (cf. [39, p. 374]) to connect the confluent hypergeometric
Ψ−function with complex parameters and real argument (occurring
in the definition of the Whittaker W-function) to the same function with com-
plex argument but real parameters:
(2x) s− 1 2 e −x Ψ(s−l,2s;2x) = 1
√ π
∞
0
e −xcosht
Ψ −l; 1
2 ;ξ
cosh s − 1
t dt,
2
where ξ = x(1+cosht). The Whittaker W-function, Wl,iR(x), is the decreasing
(as x → ∞) solution to Whittaker’s differential equation
w ′′
+ − 1 l
+
4 x +
1
4 + R2
x2
w = 0.
141

We can express Wl,iR in terms of the Tricomi confluent hypergeometric function,
Ψ, as (cf. [29, vol I, p. 264 (2)])
Wl,iR(2x) = e −x (2x) 1 2 +iR
1
Ψ + iR − l, 1 + 2iR;2x .
2
With s = 1
2
+ iR we thus get the following integral representation:
Wl,iR(2x) =
−l; 1
;x(1 + cosht) cosh[iRt]dt.
2
2x ∞
e
π 0
−xcosht Ψ
Hence the integral we want to evaluate is:
∞
I = e
0
−xcosht
Ψ −l; 1
;x(1 + cosht) cos(Rt)dt
2
= 1
∞
e
2
−xcosht
Ψ −l; 1
;x(1 + cosht) e
2 iRt dt,
that is we define
−∞
I = 1
∞
e
2 −∞
φ(t) ψ(t)dt, (4.1)
where φ(t) = −xcosht + iRt and ψ(t) = Ψ −l; 1
2 ;x(1 + cosht) .
The trick is now to turn the contour out in the complex plane into a path of
“almost stationary” phase. The argument of ψ(t) is slowly varying compared
to the argument of e φ(t) , hence a path of stationary phase for e φ(t) should
provide a good approximation to the real path of stationary phase.
The same integral, but with ψ(t) = e At has been studied extensively by
Helen Avelin [10] in connection with the K-Bessel function and her notes
have been very helpful.
We now have to find the path of stationary phase for e φ(t) ; it will depend on
the parameters x and R, or actually (as we will see) on the quotient
4.3 The path
T = R
x .
Along the desired path, the imaginary part of φ should be constant, and it
should also pass through the point where the real part of φ has a maximum.
This maximum is attained at the solutions of
142
φ ′ (t) = −xsinht + iR = 0, (4.2)

which are the following points (depending on T ):
t1 = ln(T +
T ≥ 1 :
√ T 2 − 1) + i π
2 ,
t2 = −ln(T + √ T 2 − 1) + i π
2 ,
T ≤ 1 : t = iarcsinT.
We have to separate these two cases (T ≥ 1 and T ≤ 1), and for numerical
reasons we will also treat the case of T ≈ 1 separately. We now view t as a
complex variable, t = u + iv, and thus
φ(t) = φ(u + iv)
= −xcosh(u + iv) + iR(u + iv)
= −x(coshucosv + isinhusinv) − Rv + iRu
= −[xcoshucosv + Rv] − i[xsinhusinv − Ru].
The desired path is determined by the equation
ℑφ = Ru − sinhusinv = [constant], (4.3)
where the constant must be the value of ℑφ at the solutions of (4.2). The
path can be parametrized by u, hence we will sometimes use the notation
φ(u) = φ(u + iv(u)) and ψ(u) = ψ(u + iv(u)). In the following sections we
will detail the paths for the different cases of T .
T ≥ 1 :
There are two solutions of (4.2): t1 = u1 + iv1 and t2 = −u1 + iv1, where u1 =
ln(T + √ T 2 − 1) and v1 = π
2 . Hence we must split the path into two parts, but
for numerical reasons we will actually split it into three parts.
Consider first the critical point t1 = u1 + iv1. From (4.3) we see that
ℑφ(u + iv) = ℑφ(v1)
⇔
Ru − xsinhusinv = Ru1 − xsinhu1 sinv1
= Ru1 − xsinhu1
= Ru1 − x T 2 − 1,
which, with S = Tu1 − √ T 2 − 1 = T ln(T + √ T 2 − 1)− √ T 2 − 1 is equivalent
to
Tu − S
sinv =
sinhu .
143

This equation defines the path that goes through t1, and similarly the equation
sinv =
Tu + S
sinhu
determines the path through t2. We have to be careful now to make sure we get
the correct branch of arcsinv. It turns out that we must make different choices
for u < u1 and u > u1. A similar formula will hold for negative u.
Set u0 = S
T , and ∆ = u1 − u0. It can be shown that sinv is decreasing in
[u0,∞) and we cut off the integral at u0 and use the constant value of v = v0 = π
between [−u0,u0]. Furthermore, the bulk part of the integral seems to be
roughly contained in the interval [u1 − ∆,u1 + ∆] (from numerical experiments).
We thus have five intervals:
]∞,−u1 − ∆], [−u1 − ∆,−u0], [−u0,u0], [u0,u1 + ∆], [u1 + ∆,∞[,
which will be treated separately. In the following sections arcsin always de-
note the principal branch of arcsine, i.e. arcsinx ∈ [− π π
2 , 2 ].
In all of the intervals we need to find expressions in terms of u for the
functions v, dv
du , φ(u) and cosh(u + iv(u)).
The interval u > u1
Here we have
and
dv
du
v = arcsin
Tu − S
,
sinhu
T sinhu − (Tu − S)coshu
=
sinhu sinh 2 u − (Tu − S) 2
.
It is also clear that cosh(u + iv) = coshucosv + isinhusinv = coshucosv +
i[Tu − S], hence
The interval u < u1
φ(u + iv) = −xcoshucosv − Rv − i[sinhusinv − Ru]
= −xcoshucosv − Rv − i[x(Tu − S) − Ru]
= −xcoshucosv − Rv + ixS.
Note that v1 = π
2 so in this interval we have to change the branch of arcsine,
i.e.
Tu − S
v = π − arcsin ,
sinhu
144

and
dv
du
T sinhu − (Tu − S)coshu
= −
sinhu sinh 2 .
u − (Tu − S) 2
As above, we also have cosh(u + iv) = coshucosv + i[Tu − S], and
The interval −u0 ≤ u ≤ u0
φ(u) = −xcoshucosv − Rv + ixS.
In this interval we keep v = v0 = π constant, hence
and
Here cosh(u + iv) = −coshu, and
v = π,
dv
= 0.
du
φ(u) = −xcoshucosπ − Rπ + iRu = −Rπ + xcoshu + iRu.
The interval −u1 ≤ u ≤ u0
Since we have π
2 ≤ v ≤ π here also we have to stick to the same branch of
arcsine, hence
Tu + S
v = π − arcsin ,
sinhu
and
dv
du
T sinhu − (Tu + S)coshu
=
sinhu sinh 2 .
u − (Tu + S) 2
We get cosh(u + iv) = coshucosv + i[Tu + S], and
φ(u+iv) = −xcoshucosv−Rv−i[x(Tu+S)−Ru] = −xcoshucosv−Rv−ixS.
The interval u ≤ −u1
Now v ≤ π
2 and we can get back to the principal branch:
and
dv
du
v = arcsin
Tu + S
,
sinhu
T sinhu − (Tu + S)coshu
= −
sinhu sinh 2 .
u − (Tu + S) 2
145

We have cosh(u + iv) = coshucosv + i[Tu + S], and
T ≤ 1 :
φ(u + iv) = −xcoshucosv − Rv − i[xsinhusinv − Ru]
= −xcoshucosv − Rv − i[x(Tu + S) − Ru]
= −xcoshucosv − Rv − ixS.
If T ≤ 1 we define T = R
x = sinα, and we can see that φ ′ (t) = 0⇔ t = iα, that
is, the stationary point is on the imaginary axis. We also see that φ ′′ (iα) =
−xcosh(iα) = −xcosα = −x √ 1 − T 2 = − √ x 2 − R 2 ≤ 0. Since the point t =
iα is a maximum (i.e. φ ′′ (iα) < 0) we see that there might be trouble as
T → 1; numerically it turns out that a safe restriction is
|R − x| > 5 2 3 R 1 3 ,
and for |R − x| < 5 2 3 R 1 3 we will use the path in the next section instead. The
path of stationary phase that passes through iα is given by ℑφ(u + iv) =
ℑφ(iα) ⇔ Ru − xsinhusinv = 0, that is
and
dv
du
sinv = Ru Tu
=
xsinh sinhu ,
sinhu − ucoshu
= ±R
sinhu (xsinhu − Ru)(xsinhu + Ru) ,
where +is for u > 0 and − for u < 0. We also have
and
The case T ≈ 1 :
cosh(u + iv) = coshucosv + iTu,
φ(u) = iR(u + iv) − xcoshucosv
= −xcoshucosv − Rv + i[−xsinhusinv + Ru]
= −xcoshucosv.
This case is considered separately because of the numerical reasons hinted at
above, and we will treat T close to 1 as if exactly 1. If we didn’t do that we
might for example end up with sinv > 1. The test used to indicate that T is
close to 1 is that
|x − R| < min( R
3 ,52 3 R 1 3 ).
146

We get
and as before
and
sinv = u
sinhu ,
dv
du =
sinhu − ucoshu
sinhu (sinhu − u)(sinhu + u) ,
φ(u) = −xcoshucosv − Rv + i[−xsinhusinv + Ru]
= −xcoshucosv − Rv + iu[R − x].
We also get cosh(u + iv) = coshucosv + iu.
4.4 The integrand
When we review the previous sections, we see directly that in all cases we
have v(−u) = v(u), dv dv
du (−u) = du (u) and φ(−u) = φ(−u). Using
cosh(−u + iv) = cosh(u + iv),
and the fact that we see that Ψ is analytic in its third argument we have
ψ(−u) =Ψ −l, 1
,x(1 + cosh(−u + iv(−u)) = Ψ −l,
2 1
,x(1 + cosh(u + iv)
2
=ψ(u).
Note that when using this formula we must be careful since Ψ is also in general
multi-valued in its third argument (e.g. Ψ − 1 1
2 , 2 ,z = √ z). It is customary to
use the principal branch, i.e. with a branch-cut along the negative real-axis,
but in the case T ≥ 1 however, the argument x(1 + cosh(−u + iv(−u)) passes
through the negative real axis at u = ±u0 and we must make sure we get the
correct branch choice.
We thus have the following reflectional symmetries:
v(−u) = v(u),
dv
(−u)
du
=
dv
du ,
φ(−u) = φ(u),
ψ(−u) = ψ(u),
and using these we can rewrite the complex integral (4.1) as a real integral. To
147

simplify the following equations we introduce the notation
where
We now have
I = 1
2
= 1
2
= 1
2
= 1
2
=
=
∞
−∞
∞
−∞
0
−∞
∞
0
∞
0
∞
0
e φ(u) ψ(u) = P(u) = R(u)e iΘ(u) ,
Θ(u) = ℑφ(u) + Argψ(u), and
R(u) = |ψ(u)|e ℜφ(u) .
e φ(t) ψ(t)dt
P(u) 1 + i dv
du
du
P(u) 1 + i dv
du +
du
1
∞
P(u) 1 + i
2 0
dv
du
du
P(−u) 1 − i dv
+ P(u) 1 + i
du
dv
du
du
1
i
dv
(P(u) + P(−u)) + (P(u) − P(−u))
2 2 du du
R(u) cos(Θ(u)) − sin(Θ(u)) dv
du,
du
which is now a completely real integral. When we split it up for the different
cases we get explicit expressions for the various factors.
T ≥ 1 :
Here we have
where now
and
148
I1 =
u0
0
= e −Rπ
I =
u0
0
= I1 + I2,
∞
+
u0
R(u) cos(Θ(u)) − sin(Θ(u)) dv
du
du
u0
0
e xcoshu |ψ(u)|{cos(Ru + Argψ(u))}du,
∞
I2 = R(u) cos(Θ(u)) − sin(Θ(u))
u0
dv
du,
du

where
Θ(u) = xS + Argψ(u),
T ≤ 1 :
Here
T ≈ 1 :
I =
=
=
−∞
∞
∞
−∞
∞
0
∞
0
P(u)
R(u)
e φ(u) |ψ(u)|
R(u) = |ψ(u)|e −xcoshucosv−Rv .
1 + i dv
du
The relevant integral is given by
∞
I = P(u) 1 + i dv
du =
du
=
=
0
∞
0
R(u)
e φ(u) |ψ(u)|
du =
cos(Θ(u)) − sin(Θ(u)) dv
du
du.
cos(Argψ(u)) − sin(Argψ(u)) dv
du.
du
cos(Θ(u)) − sin(Θ(u)) dv
du
du.
cos(uδ + Argψ(u)) − sin(uδ + Argψ(u)) dv
du.
du
These integrals we have obtained can now be dealt with using standard
methods for numerical integration, e.g. Gaussian quadrature once we have
satisfying algorithms to compute the function ψ.
T ≥ 1 :
Here we have
where now
I1 =
u0
0
= e −Rπ
I =
u0
0
= I1 + I2,
∞
+
u0
R(u) cos(Θ(u)) − sin(Θ(u)) dv
du
du
u0
0
e xcoshu |ψ(u)|{cos(Ru + Argψ(u))}du,
149

and
∞
I2 = R(u) cos(Θ(u)) − sin(Θ(u))
u0
dv
du,
du
where
Θ(u) = xS + Argψ(u),
T ≤ 1 :
Here
T ≈ 1 :
I =
=
=
−∞
∞
∞
−∞
∞
0
∞
0
P(u)
R(u)
e φ(u) |ψ(u)|
R(u) = |ψ(u)|e −xcoshucosv−Rv .
1 + i dv
du
The relevant integral is given by
∞
I = P(u) 1 + i dv
du =
du
=
=
0
∞
0
R(u)
e φ(u) |ψ(u)|
du =
cos(Θ(u)) − sin(Θ(u)) dv
du
du.
cos(Argψ(u)) − sin(Argψ(u)) dv
du.
du
cos(Θ(u)) − sin(Θ(u)) dv
du
du.
cos(uδ + Argψ(u)) − sin(uδ + Argψ(u)) dv
du.
du
These integrals we have obtained can now be dealt with using standard
methods for numerical integration, e.g. Gaussian quadrature once we have
satisfying algorithms to compute the function ψ.
4.5 Evaluation of the Integrand
As a final step to be able to evaluate the integrals in the previous section we
need efficient algorithms for the function Ψ −l; 1
2 ,z where l is real and z is
a complex number. In most papers on numerical algorithms for Ψ(a,c,z) the
values of z are restricted to the reals, cf. e.g. [96]. Algorithms without this
restriction can be found in e.g. [102].
150

In this chapter we will outline the three different approaches which seems
to yield the most efficient algorithms in different z-domains. The methods
include power-series for small |z|, backward-recursion for middle range |z|
and an asymptotic series expansion for large |z|. Each of them is evaluated to
at least 12-14 significant digits in the relevant interval.
It should also be noted that for the special case when l = 1
2 (which will be
used later) we have
Ψ − 1 1
,
2 2 ,z
= √ z,
and another special case is (of course) weight 0, that is l = 0, where we have
for all c
Ψ[0,c,z] = Φ[0,c,z] = 1.
These special cases can be easily deduced from the power series expression in
the next subsection.
4.5.1 The power series
It is known that we can express Ψ as a sum of two Φ functions (cf. [99, p. 20]
or [29, p. 257]):
Ψ(a,c,z) =
Γ(1 − c)
Γ(c − 1)
Φ(a,c,z) + z
Γ(a − c + 1) Γ(a)
1−c Φ(a − c + 1,2 − c,z),
hence we actually only need an algorithm for Φ (the algorithm for Γ, using
Stirling’s formula is standard). We know that Φ has a power-series expansion
valid for all a, z and c not a negative integer (if c is a negative integer it is valid
if a = c)
Φ(a,c,z) =
∞
∑
n=0
(a)n
where (a)n is the Pochammer symbol, defined as (a)k = a(a+1)···(a+k−1),
for a complex (not a negative integer > −k +1) and k a positive integer. It can
also be written as (a)k = Γ(a+k)
Γ(a) . We see that for non-exceptional a and c (i.e.
c not very close to a negative integer) the power series should converge in a
similar manner as the power series for the exponential function. A few special
(c)n
z n
n! ,
151

cases worth mentioning are
Φ(0,c,z) = 1,
Φ(a,a,z) = e z ,
Φ(−1, 1
,z)
2
Φ 1,
= 1 − 2z,
1
2 ,z
= √ π
∞
∑
n=0
z n
Γ 1
2 + n.
To evaluate the power series one can use for example a direct formula, recursion,
or a continued-fraction method (c.f. [71, p. 185]). We set
A−1 = A0 = 1, A1 = 1 + az
c ,
ri =
(a + i − 1)
, i > 1,
i(c + i − 1)
Ai = Ai−1 + (Ai−1 − Ai−2)riz, for i > 1.
Then the i − thapproximation to Φ(a,c,z) is given by Ai, and we quit the
summation whenever |Ai − Ai−1| < ε.
The special case of c = 1
2
Since what we in general want to evaluate is Ψ(a, 1
2 ,z) we can formulate the
specific formula for this case,
Ψ a, 1
2 ,z
Γ
=
1
2
Γ a + 1
Φ a,
2
1
2 ,z
+ Γ− 1
2 √
zΦ a +
Γ(a)
1 3
,
2 2 ,z
Γ
=
1
2
Γ a + 1
∞ (a)n
∑
1
2 n=0 2 n n!zn + Γ− 1
2 √ ∞ 1 a + 2
z
Γ(a) ∑ n
zn
3
n=0 2 n! n
Γ
=
1
2
Γ a + 1
∞ Γ(a + n)Γ
∑
2 n=0
1
2
Γ(a)Γ 1
2 + nΓ(n + 1) zn
+ Γ− 1
2 √ ∞ Γ
z
Γ(a) ∑
n=0
a + 1
2 + nΓ
3
2
Γ a + 1
3
2 Γ 2 + nΓ(n + 1) zn
Γ
=
1 2
2
Γ a + 1
∞
∑ Γ(a + n) −
2 Γ(a) n=0
√ z Γa + 1
2 + n
×
1
2 + n
×
z n
Γ(n + 1)Γ n + 1
.
2
This sum can now be evaluated by any standard method. Define
152
an =
Γ(a + n)
Γ(n + 1)Γ n + 1
, bn =
2
Γ a + 1
2
Γ(n + 1)Γ n + 1
2
+ n
1
2 + n,

and compute the sum
∞
∑ an −
n=0
√ n
zbn z .
Since the an and bn are independent of z and this sum is evaluated many times
it is preferable to make a table of an and bn. These can either be computed
by recursion to get good speed, or since we need only tabulate them once by
direct evaluation.
Reflection
The power series is obviously valid in a circular region in the plane, but experimentally
it seems more efficient to use Kummer’s transformation for values
of z in the left half-plane:
Φ(a,c,z) = e z Φ(c − a,c,−z),
which gives the following formula for Ψ:
Ψ(a,c,−z) =
e −z
Γ(1 − c)
Γ(c − 1)
Φ(c − a,c,z) + (−z)1−c Φ(1 − a,2 − c,z) ,
Γ(a − c + 1) Γ(a)
where Φ can be evaluated as above.
4.5.2 Recursion
It is well-known that both confluent hypergeometric functions satisfy many
recursion relations (cf. [29, pp. 254, 257-258]), and one can use these to
formulate recurrence equations, cf. e.g. [96, pp. 64-65]. If we set un =
(a)nΨ(a + n,c,z) we get the following equation:
(a + n − 1)un−1 + (c − 2a − z − 2n)un + (a + n + 1 − c)un+1 = 0, (4.4)
which is valid for all values of a,c,z and for all integers n ≥ 1. What we want
to compute is now u0, the first term in the recursion, hence we can’t use regular
forward-recursion. What we have to do is basically to start with some values
of uN and uN−1 and then go backwards. The standard way to do this is by a so
called Miller algorithm (cf. e.g. [32, 73, 74] or in this particular case [96] and
[102]). In order to use this kind of algorithm we need a normalization, and in
this case we use
S =
∞
∑ λnun = z
n=0
−a ,
153

where λn = (−1) n
c − a − 1
n
= (−1) n Γ(c−a)
Γ(c−a−n)n! . This follows from the
binomial theorem for general exponents. What we do next is to rewrite the
recursion equation (4.4) as
where
an =
un+1 + anun + bnun−1 = 0,
c − 2a − z − 2n
1 + n + a − c
and bn =
a + n − 1
1 + n + a − c .
There are several choices in the implementation and the one we use is basically
[32, p. 37]. This is actually a continued-fraction method for the quotients
un+1/un. For a fixed positive integer N > 0 we set
and then recursively set
s (N)
N
r (N)
n−1 =
= r(N)
N
−bn
an + rn
= 0,
, and
s (N)
n−1 = rn−1(λn + s (N)
n ),
for n = N,N − 1,N − 2,...1. In the final step we get an approximation to u0:
u (N)
0 =
S
λ0 + s (N)
0
To get an approximation to u0 with an error smaller than ε we choose a starting
integer M and repeat the process for the sequence of N = M,M +1,... and stop
when the difference |u (N)
0 − u(N+1)
0 | < ε. To avoid unnecessary computations
the choice of M is important, and should be tabulated in advance.
Note that λn satisfies a recursion relation: λ0 = 1, and λn = − c−a−n
n λn−1,
and if we compute {λ j} M 1 in advance we only need to compute an additional
λN in each step.
From the formula for an and bn we see that the only points we should avoid
are those where 1 + a − c is a negative integer. Hence this method should be
used for the sake of precision when c is close to an integer but 1 + a − c is not
a negative integer.
154
.

4.5.3 Asymptotic series
The following asymptotic expansion for Ψ is well-known (see [29, p. 278])
Ψ(a,c,z) =z −a
N
∑ (−1)
n=0
n (a)n(a − c + 1)n
n!zn + O(z −N−1
) , as |z| → ∞,
for |Argz| ≤ 3π
2 ,
and when evaluated by forward-recursion on the coefficients the convergence
is fast for large |z|. As is usual when evaluating asymptotic series we sum
the series up to the term N = N0 where the terms starts to increase in absolute
value.
4.5.4 Comparison
We made a spot-check of the methods for computing Ψ(−l, 1
2
,x + iy) over
a grid where |l| ≤ 1 and 1 ≤ x ≤ 100 and for each (l,x) we let y vary between
1 and lnx to simulate the behaviour of cosh(x + iy) at the paths above,
where basically the imaginary part is sinvsinhu = O(u) and the real part is
cosvcoshu = O(eu ). In this region we first of all compared the three methods
with each other with respect to speed and we also made spot-check against
Maple and PARI (with internal precision set to 100 decimal places). We found
that the program was accurate up to 12-13 significant digits in the entire region
when we choose the algorithms as follows:
0 ≤ x ≤ 25 Power series
25 ≤ x ≤ 35 Recursion
35 ≤ x Asymptotic series
It should be noted that the recursive algorithm actually provides 12-13 decimal
places (depending on the termination condition of course) for x ≥ 0.1, but is
in general very slow for small x. By further sporadic spot-checking we also
drew the conclusion that the Asymptotic series is accurate up to at least 12-
13 decimal places when |z| ≥ 35. Negative values of x was checked against
positive values by the reflection relation stated above that relates Ψ(a,c,−x)
with Ψ(a,c,x).
4.6 Evaluation of the Integrals
The infinite integrals I were first truncated to finite integrals using the fact that
,x) grows at most like a polynomial in x of degree ≤ |l| and that eℜφ
Ψ(−l, 1
2
155

is decreasing like e−ex. To obtain the bound on Ψ(−l, 1
2 ,x) study the integral
representation
Ψ(−l, 1
1 ∞
,x) = e
2 Γ(−l) 0
−xt t −l−1 (1 +t) l− 1 2 dt,
do a variable substitution u = xt and compare with the integral representation
and known bounds for the Γ function.
The actual bounds we use are verified by the numerical experiments. The
finite interval is then divided up in to smaller intervals (according to the value
of R) and on each of the smaller intervals we evaluate the integral using a
16-point Gaussian quadrature rule.
4.7 Remarks on the K-Bessel function
We know that the special case l = 0 in Wl,iR(x) gives us essentially the K-
Bessel function, KiR(x) (i.e. W0,iR(2x) =
2x
π KiR(x)). The algorithm de-
scribed in this paper is then identical to Hejhal’s algorithm ([42]), generalized
by Avelin to Ks(x) for complex s ([9]).
Note that
KiR(x) =
π
2x e−x (2x) 1 2 +iR
1
Ψ + iR,1 + 2iR;2x ,
2
but even though the arguments to the Ψ−function are complex it turns out that
in this special case most of the algorithms we used to compute the Ψ-function
with real arguments can be trivially modified and actually show good convergence
properties. Hence we can use both the backwards recursion (Millertype)
algorithm and the power series to compute the K-Bessel function.
One particularly nice power series method for K-Bessel function is the following.
From the expression of KiR as a sum of IrR and I−iR it is easy to see the
power series defining KiR from the known power series for IiR. From [33, pp.
399-400], and breaking out the dominating factors from the expansion we see
that
with
156
1 π
KiR(x) =
|Γ(1 + iR)| 2sinh(πR)
fk = −i|Γ(1 + iR)|
∞
∑ ck fk
k=0
x −iR
2
Γ(k + 1 − iR) −
x iR
2
Γ(k + 1 + iR)

and
and
rk = R|Γ(1 + iR)|
x −iR
2
Γ(k + 1 − iR) +
x iR
2
Γ(k + 1 + iR)
,
ck = 1
k!
x 2
4
k
.
The recursive relations are not affected by this scaling and are still
fk = k fk−1 + rk−1
k2 + R2 rk =
1
k2 + R2 [(2k − 1)rk−1 − rk−2]
ck = x2 ck−1
4 k
for k ≥ 1 where the initial data now are
f0 = −2sinθ,
r0 = 2Rcosθ, and
r1 =
R
(cosθ + Rsinθ)
1 + R2 with θ = Rln( x
2 ) − ArgΓ(1 + iR). Now we can just compute the sum by standard
forward-recursion and multiply with the Γ and sinh factors at the end.
(And for large R avoid cancellation and division by infinity by simply take
both factors into account simultaneously). (See also [98, pp. 377-379].)
4.8 Numerical verification
The numerical verification of the algorithm has been done by comparing the
values of Wl,iR(x) against Maple with 25 digits precision. The comparison has
been done over 2 separate grids in the (R,x,l) space:
1. 200000 pseudo random points (Ri,xi,li) in the cube
[0,20] × [0,20] × [0,6], and
2. 100000 pseudo random points (Ri,xi,li) in the cube
[20,100] × [0,20] × [0,6].
The reason for the choice of [0,6] as the l-interval is because that is the range
we will use in the application of this algorithm to the problem of computing
Maass waveforms with nonzero weight.
157

We consider the relative errors
εi = Wli,iRi (xi) Fortran −Wli,iRi (xi) Maple
Wli,iRi (xi) Maple −1
as corresponding to a set of (pseudo)random variables. As such we consider
their mean and variance, apart from the obvious study of their absolute maximum.
We also consider the mean and standard deviance of the absolute values
|εi|.
1. In the first grid we have
2. In the second grid we have
max{|εi|} = 5.3E − 10,
mean{εi} = −2.0E − 15,
mean{|εi|} = 4.2E − 14,
std{|εi|} = 3.5E − 12,
std{εi} = 3.5E − 12.
max{|εi|} = 1.7E − 09,
mean{εi} = 2.2E − 14,
mean{|εi|} = 8.1E − 14,
std{|εi|} = 5.5E − 12,
std{εi} = 5.5E − 12.
In connection with this it might also be enlightening to know that CPU time
required for Maple to make the computations over the first grid is approximately
27 hours, where my Fortran program only need about 15 minutes (on
a 1533MHz CPU). Over the second grid the needed CPU times are approximately
25h45m for Maple and 33m for Fortran (on a 1533MHz CPU). These
figures are not exact since there were other programs running simultaneously
on this computer too, but they serve well as a hint of the effectiveness of the
algorithm.
158
,

Swedish Summary
Maassvågfunktioner ur ett beräkningsperspektiv
Detta är en avhandling i experimentell matematik, och vi befinner oss således
i en situation där den “rena” matematiken drar nytta av numeriska resultat för
att bygga hypoteser.
En kortfattad beskrivning av vad som kännetecknar experimentell matematik
är att man med hjälp av teoretiska resonemang konstruerar numeriska
modeller (algoritmer) vilka sedan används för att generera information. Utifrån
de numeriska resultaten bygger man sedan upp hypoteser och förmodanden
(för en mer detaljerad beskrivning se [16]).
Förhoppningen är naturligtvis att hypoteserna sedan skall kunna bevisas
strikt matematiskt, men även i de fall då det nuvarande matematiska maskineriet
inte förmår detta kan de numeriska resultaten ge värdefulla insikter
och inspirera till fortsatta studier.
Avhandlingens tre huvudkapitel är alla utformade på samma sätt; merparten
ägnas åt att presentera den teori som sedan används för att konstruera algoritmer.
Efter en beskrivning av hur algoritmerna kan utformas presenteras de
experimentella resultat vi kommit fram till.
Det problem vi studerat är att konstruera de matematiska objekt som kallas
“Maassvågfunktioner” (efter Hans Maass (1911-1992)). Matematiska tillämpningar
av dessa vågfunktioner finns främst inom analytisk talteori. Det finns
även fysikaliska kopplingar, främst då till matematiska modeller av kvantkaos
([35, 14]), men även till kosmologi ([8]).
Något om den fysikaliska bakgrunden
Betrakta en (kvantmekanisk) partikel med massa m0 som rör sig fritt på en
yta, S . Enligt kvantmekaniken beskrivs detta system av en vågfunktion, ψ,
vilken talar om för oss hur stor sannolikheten är att påträffa partikeln i ett visst
område. Vågfunktionens utbredning, för en partikel med energin E = ¯hv, ges
i sin tur av Schrödingerekvationen:
i¯h ∂ψ
∂t
¯h2
= − ∆ψ = Eψ,
2m0
159

där ∆ är den så kallade Laplace-Beltramioperatorn på ytan. (T.ex. ges den i
planet av ∆ψ = ψ ′′
xx + ψ ′′
yy). Vi antar nu att det går att skriva ψ som en produkt
av en tidsberoende del, e−ivt , och en tidsoberoende del, φ(P), P ∈ S . Vi
får då en ekvation för den tidsoberoende delen (den tidsoberoende Schrödingerekvationen,
eller Laplaces ekvation):
∆φ + λφ = 0, (⋆)
där vi har satt λ = 2m0E
¯h 2 (egenvärdet), och eftersom sannolikheten för att partikeln
befinner sig på ytan är ett måste vi dessutom ha
|φ| 2 dP = 1.
S
Om partikeln är laddad och vi kopplar på ett konstant magnetiskt fält, B, på
ytan kommer den tidsoberoende delen av vågfunktionen att få ett “tvist” (spin)
och φ måste nu uppfylla en variant av Laplaces ekvation:
∆Bφ + λφ = 0, (⋆⋆)
där ∆B är en modifierad Laplaceoperator, (i vårt fall blir den ∆B = ∆ − iyB ∂
∂x
med B proportionell mot den magnetiska fältstyrkan).
Observera att vågekvationen, (⋆) är “universell”, i den meningen att den
beskriver många olika former av vågrörelse. T.ex. om S får beskriva ett
trumskinn, så kommer (⋆) att beskriva utseendet hos en stående våg på trumskinnet,
med en frekvens proportionell mot √ λ. Bastoner svarar alltså mot
små värden på λ och ljusare toner mot högre värden på λ.
I den här avhandlingen betraktar vi ekvationerna (⋆) och (⋆⋆) för vissa
speciella ytor S , nämligen sådana med konstant negativ krökning (vi säger
att ytan är hyperbolisk), ändlig area samt ett ändligt (positivt) antal “punkteringar”
eller “spetsar” (dvs vi har plockat bort ett ändligt antal punkter från
ytan och “dragit ut dem” till oändligheten). Dessutom måste ytorna vara
speciella på ett “aritmetiskt” sätt, vilket i det här fallet betyder att de ges av en
delgrupp till den modulära gruppen. Vi tillåter också ytor med en sorts “hörn”.
Lösningarna till (⋆) och (⋆⋆) på sådana ytor kallas för Maassvågfunktioner.
En orsak till att hyperboliska ytor, är intressanta i detta sammanhang är
att den klassiska dynamiken för en fri partikel på sådana är (starkt) kaotisk.
Genom att studera lösningarna till (⋆) på en sådan yta kan vi sålunda komma
till insikt om det finns “spår” av detta klassiska kaos kvarlämnat i det kvantiserade
systemet. Denna typ av undersökningar är en del av det forskningsområde
som kallas kvantkaos. Då vågfunktionerna i sig inte är kaotiska visar sig
det klassiska kaoset i bl.a. energinivåernas (egenvärdenas) fördelning samt
utseendet hos de vågfunktioner φ som svarar mot höga energier. I Figur 2
ser du ett exempel på en Maassvågfunktion, ϕ med ett egenvärde λ ≈ 10000.
160

(a) visar nodlinjerna (dvs. de punkter där ϕ(z) = 0), och i (b) kan du se sannolikhetsfunktionen
som associeras till ϕ, dvs |ϕ(z)| 2 . De ljusa områdena i
(b) visar de platser i planet där den kvantmekaniska partikeln gärna hänger
omkring.
Det huvudsakliga syftet med arbetet som ligger till grund för denna avhandling
var att ta fram stabila algoritmer för att beräkna egenvärden, λ, samt
motsvarande vågfunktioner, φ, till ovanstående problem.
Kort om Hyperbolisk Geometri
För att åskådliggöra negativ krökning (hyperbolisk geometri) brukar man som
matematiker arbeta i t.ex. övrehalvplansmodellen, dvs
H = {x + iy ∈ C|y > 0},
och där införa ett avstånds- samt areabegrepp med hjälp av bågelementet
och areaelementet
ds = |dz|
y ,
dµ = dxdy
.
y2 Den kortaste vägen mellan två punkter z1,z2 ∈ H får man genom att följa
den geodet som passerar genom dessa. I vanliga fall (dvs då vi inte har någon
krökning) så är geodeten den unika räta linje som går mellan punkterna. I det
hyperboliska övre halvplanet ges geodeterna istället av halvcirklar samt räta
linjer (vilka kan betraktas som generaliserade halvcirklar), vilka står vinkelrätt
mot x-axeln. Se Figur S1 för en illustration av geodeterna γ1 och γ2 som
Figure S1: Övre halvplanet
H
γ3
y
z4
z3
γ1
γ2
z2
z1
(a) Geodeter och en horocykel
x
S
F1 F4 F7
E
e2
e3 e ′ 3
F6
F3
F5
F2
(b) Fundamentalområde
förbinder z1 med z2 respektive z3 med z4. I figuren ger vi även ett exempel på
161

en så kallad horocykel, γ3.
En konsekvens av detta sätt att mäta på i det övre halvplanet är t.ex. att man
inte kan ta sig ända ner till den reella linjen (dvs avståndet från varje punkt i
H till reella linjen är oändligt).
Ett något mer fysikaliskt sätt att tänka sig detta hyperboliska rummet på är
följande (fritt efter Poincaré [82]):
"Tänk dig att du lever i ett sfäriskt universum begränsad av ett “skal”. Antag att
temperaturen i detta universum är proportionellt mot avståndet till skalet, dvs
det är varmast i mitten och på skalet råder den absoluta nollpunkten. Föreställ
dig nu att du rör dig med en hastighet proportionell mot temperaturen (detta
är vad man fysikaliskt förväntar sig av på molekylär/atomär nivå). Dvs, ju
närmare skalet man kommer, desto kallare blir det, desto saktare färdas man.
Det är då ganska enkelt att inse att om man bestämmer sig för att försöka åka till
“universums ände” (skalet) så kommer man aldrig fram hur länge man än åker
(inom ändlig tid). Matematiskt sett kan man beräkna att r(t) = R(1 − e −ct ),
där r(t) är avståndet från centrum vid tiden t, R är radien på skalet och c är en
positiv konstant."
En isometri i ett rum (där vi har ett avståndsbegrepp) är en avbildning som
bevarar avstånd och areor. I vanliga “okrökta” fallet är translationer, rotationer
och speglingar exempel på isometrier. Spegling är ett exempel på en
orienteringsreverserande isometri, medan translation och rotation är orienteringsbevarande.
I det övre hyperboliska övre halvplanet så kan varje orienteringsbevarande
isometri skrivas som en Möbiusavbildning:
T : z ↦→
az + b
, där a,b,c,d ∈ R, ad − bc = 1.
cz + d
En sådan avbildning associeras sedan till ett par av matriser:
±
a
c
b
d
,
och gruppen av alla sådana isometrier skriver vi oftast som PSL(2,R) =gruppen
av alla matriser
a b
c d , med a,b,c,d ∈ R, ad − bc = 1, och där vi identifierar
matriserna
a b −a −b
c d och −c −d .
Man vet nu att alla hyperboliska ytor kan beskrivas med hjälp av diskreta
undergrupper till PSL(2,R), dvs vi kan skriva vår yta S som S = Γ\H , där
Γ är en Fuchsisk grupp (en diskret grupp av Möbiusavbildningar). De ytor, S ,
vi är intresserade av har dels “horn” (punkter utdragna till oändligheten, och
vi tillåter dessutom att de har “hörn”. I den motsvarande gruppen, Γ, visar
sig hornen som fixpunkter till paraboliska avbildningar, dvs de svarar mot
matriser med absolutbeloppet av spåret = 2 (kom ihåg att spåret av en matris
162

a b
c d definieras som a + d). Hörnen är å andra sidan fixpunkter till elliptiska
avbildningar (|spåret| < 2). Man brukar därför även säga att punkterna är
paraboliska respektive elliptiska. (De paraboliska fixpunkterna kallas också
spetsar). I PSL(2,R) finns förutom elliptiska och paraboliska avbildningar
(som geometriskt svarar mot rotationer och translationer) även hyperboliska
(|spåret| > 2). Då de andra typerna har unika fixpunkter har hyperboliska
avbildningar i regel två fixpunkter i R ∪ {∞} (och de beskriver geometriskt en
förflyttning (skalning) längs den geodet som förbinder dessa).
För att illustrera ytan S = Γ\H brukar man ta hjälp av ett fundamentalområde,
vilket i vårt fall, kan tas som ett sammanhängande område F ⊂ H vilket
har två egenskaper: för det första får inga punkter inuti F avbildas till varandra
med avbildningar i Γ, och för det andra så ska Γ-bilderna av F tesselera H
(dvs ∪γ∈Γγ(F ) = H). Ett exempel på fundamentalområde är standardfundamentalområdet
för den modulära gruppen, PSL(2,Z) (de matriser i PSL(2,R)
som endast innehåller heltal) F1 = {z = x + iy ∈ H||z| ≥ 1, |x| ≤ 1
2 }.
I Figur S1(b) kan vi se F1 tillsammans med 6 stycken “kopior”. Observera
att alla bilder, F j, har samma hyperboliska area. (Just den här samlingen
av kopior “råkar” också vara ett fundamentalområde till en grupp, Γ7,12, som
återkommer i kapitel 3). Om vi identifierar sidorna i F1 med avbildningarna
S : z ↦→ z + 1 och E : z ↦→ −1
z får vi (topologiskt sett) en sfär med en spets
(i∞) samt två hörn (e2 samt e3 som identifieras med e ′ 3
). Man kan visa att de
sidparandeavbildningarna E och S genererar PSL(2,Z), och faktum är att i de
fall vi betraktar går det alltid att välja fundamentalområdet så att de sidparande
avbildningarna genererar gruppen.
Sammanfattning av Kapitlen i Avhandlingen
I kapitel ett studerar vi det problem som i övrehalvplansmodellen kan skrivas
som:
y 2 φ ′′
xx(z) + φ ′′
yy(z) + λφ(z) = 0, z = x + iy ∈ H , (1)
φ (T z) = χ(T )φ(z),
T ∈ Γ, z ∈ H , (2)
|φ| 2 dµ = 1, (3)
Γ\H
där χ : Γ → S1 är en jämn Dirichlet karaktär (dvs χ(AB) = χ(A)χ(B) och
χ(−1) = 1), och Γ är en Hecke-kongruensundergrupp till den modulära gruppen.
För ett givet positivt heltal, N, definieras Hecke-kongruensgruppen,
Γ0(N), på följande sätt:
Γ0(N) =
a
c
b
d
|a,b,c,d ∈ Z, ad − bc = 1, c ≡ 0 mod N .
163

Vi beskriver kortfattat algoritmen. Givet ett egenvärde, λ = 1
4 + R2 , så vet vi
att φ kan skrivas som en Fourierserie:
φ(x + iy) = ∑ c(n)
n=0
√ yKiR(2π|n|y)e 2πinx ,
där KiR(y) är en K-Besselfunktion, och om vi approximerar funktionen φ med
en trunkerad Fourierserie kan denna inverteras som en diskret Fouriertransform.
Genom att använda (2) kan vi åstadkomma ett icke-trivialt, välkonditionerat
linjärt ekvationssystem vilket vi kan lösa för att få fram Fourierkoefficienterna
c(n).
Om vi gör detta för ett godtyckligt värde på λ kommer den resulterade
funktionen φ naturligtvis inte vara invariant under Γ. Men om λ är nära ett
riktigt egenvärde så kommer de uträknade Fourierkoefficienterna att vara nära
koefficienterna till en riktig egenfunktion. Med hjälp av vissa egenskaper som
vi vet att en genuin egenfunktion har så bildar vi en “funktional” h(R) som har
egenskapen att den är väldigt nära noll ifall R är nära ett korrekt egenvärde.
Denna funktion används sedan för att lokalisera egenvärden till önskad noggrannhet.
När vi väl har ett egenvärde och en tillhörande uppsättning Fourierkoefficienter
kan vi med hjälp av en ytterligare algoritm (Fas 2) beräkna
ytterligare Fourierkoefficienter på ett väldigt enkelt sätt.
I kapitel 2 studerar vi ett liknande problem, där vi istället för (1) har ekvationen
y 2 φ ′′
xx(z) + φ ′′
yy(z) − iymφ ′ x + λφ(z) = 0,z = x + iy ∈ H , (1’)
där vikten, m, är ett reellt tal, och där villkoret (2) ersätts med
φ(T z) = v(T ) jT (z;m)φ(z), (2’)
där v : Γ → C är ett så kallat multiplikatorsystem, och
jT (z;m) = exp(2πimArg(cz + d)),
för T =
a b
c d och m ∈ R.
Fysikaliskt kan ekvationen (1’) tolkas som att vi nu har en laddad (kvantmekanisk)
partikel som rör sig i ett konstant magnetiskt fält på ytan Γ\H. Den
magnetiska fältstyrkan är proportionell mot m, och multiplikatorsystemet kan
användas för att beskriva flödet genom olika delar av ytan (“hornen”, “handtagen”
etc.) Se t.ex. [2, 1, 11]. (För den som är insatt i fysiken så svarar (1’)
mot vi väljer Landaugaugen, A = c m
y dx för någon lämplig konstant.)
Det visar sig att den metod som infördes i kapitel 1 fortfarande fungerar,
med vissa smärre förändringar. Det (ur beräkningssynpunkt) svåraste problemet
är att Fourierutvecklingen innehåller Whittaker-W- istället för K-Bessel
funktioner. W-funktionen är svårare att beräkna, och vi var tvungna att utveckla
164

en ny algoritm för denna. Algoritmen presenteras i kapitel 4.
I kapitel 3 undersöker vi samma problem som i kapitel 1, men vi betraktar
här även undergrupper Γ ⊆ PSL(2,Z) som inte är kongruensundergrupper.
Den främsta skillnaden mellan kongruens- och icke-kongruensgrupper är
att man inte har lika många “symmetrier” tillgängliga i icke-kongruensfallet.
Detta innebär t.ex. att man inte har så mycket a priori kännedom om vare
sig spektrat eller egenfunktionerna. Man kan därför ställa sig flera frågor
som är mer eller mindre besvarade i kongruensfallet: finns det några “nya”
Maassvågfunktioner? hur många finns det i så fall? hur fördelar sig Fourierkoefficienterna?
Uppvisar de några symmmetrier?
Vi bevisar existensen av nya vågfunktioner för grupper Γ som har vissa
specifika egenskaper, vilka på ytorna Γ\H motsvarar olika former av spegelsymmetrier.
(För att systematiskt verifiera dessa egenskaper måste man dock
använda datorer i alla utom de enklaste fallen).
165

Acknowledgements
I would like to express my deep gratitude towards my supervisor, Prof. Dennis
A. Hejhal. He has been my constant guide in this exciting world of hyperbolic
geometry, quantum chaos and analytic number theory. I wouldn’t have missed
this for anything! I am also very thankful for all the work he has put in by
proofreading all my manuscripts for this thesis.
I would also like to thank my assistant advisor, Dr. Andreas Strömbergsson
for the many discussions, which have been truly invaluable. My fellow graduate
student, Helen Avelin has also contributed to many of these discussions,
and I would like to thank her too, especially in connection with the work in
Chapter 4.
The computational parts of my work would not have been possible without
the care of the computer system that has been provided by all competent
and friendly computer administrators here at the department and at NSC in
Linköping.
When widening the area of gratefulness, I would also like to thank all my
co-workers at the department for being around and providing a nice atmosphere
for me to work and drink coffee in. I mention no one in particular,
since that’s the only way I won’t forget anyone.
Close by the department, I should also thank Möbius (the student organization)
for providing me with a constant supply of cc-light for the last 12 years
or so.
Thanks to all my friends and acquaintances in Uppsala for providing nonmathematical
reasons to stay here.
While I am at it, I would also like to thank my parents and family for this
(once in a lifetime) opportunity to exist.
And, finally, I would like to thank ............... (insert your name here if you
feel left out).
167

References
[1] M. Antoine, A. Comtet, and M. Knecht, Heat kernel expansion for
fermionic billiards in an external magnetic field, J. Phys. A: Math. Gen.
23 (1990), L35–L41.
[2] M. Antoine, A. Comtet, and S. Ouvry, Scattering on a hyperbolic torus in
a constant magnetic field, J. Phys. A: Math. Gen. 23 (1990), 3699–3710.
[3] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory,
Springer-Verlag, 1976.
[4] T. Arakawa, Shimura correspondence for Maass wave forms and Selberg
zeta functions, Proceedings of the conference "automorphic forms and representations
of algebraic groups over local fields" (Hiroshi Saito and Tetusya
Takahashi, eds.), RIMS Kyoto Univ., 2003, pp. 1–14.
[5] A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(M), Math. Ann. 185
(1970), 134–160.
[6] A. O. L. Atkin and W.-C. W. Li, Twists of newforms and pseudoeigenvalues
of W-operators, Invent. Math. 48 (1978), no. 3, 221–243.
[7] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence
subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX,
Univ. California, Los Angeles, Calif., 1968), Amer. Math. Soc., Providence,
R.I., 1971, pp. 1–25.
[8] R. Aurich, F. Steiner, and H. Then, Numerical computation of Maass waveforms
and an application to cosmology, Proceedings of the "International
School on Mathematical Aspects of Quantum Chaos II", Lecture Notes in
Physics, Springer, 2004, to appear.
[9] H. Avelin, On the Deformation of Cusp Forms, 2003, Filosofie Licentiat
avhandling.
[10] , Working notes on an algorithm for the K-bessel function with
complex order, 2004, Personal Communication.
169

[11] J. E. Avron, M. Klein, and A. Pnueli, Hall Conductance and Adiabatic
Charge Transport of Leaky Tori, Physical Review Letters 69 (1992), no. 1,
128–131.
[12] E. Balslev and A. Venkov, Spectral theory of Laplacians for Hecke groups
with primitive character, Acta Math. 186 (2001), 155–217.
[13] A. Biró, Cycle integrals of Maass forms of weight 0 and fourier coefficients
of Maass forms of weight 1/2, Acta Arithmetica 94 (2000), no. 2,
103–152.
[14] E. Bogomolny et al., Arithmetical Chaos, Physics Reports 291 (1997), 219–
324.
[15] A. Booker, A. Strömbergsson, and A. Venkatesh, Effective Computations
with Maass Forms, In preparation.
[16] J. Borwein and D. Bailey, Mathematics by experiment, A K Peters Ltd.,
Natick, MA, 2004, Plausible reasoning in the 21st century.
[17] R. W. Bruggeman, Modular forms of varying weight. III, J. Reine Angew.
Math. 371 (1986), 144–190.
[18] , Families of automorphic forms, Birkhäuser, 1994.
[19] D. Bump, The Rankin-Selberg method: a survey, Number Theory, Trace
Formulas and Discrete Groups (K.E. Aubert et al, ed.), Academic Press, 1989,
pp. 49–109.
[20] , Automorphic Forms and Representations, Cambridge Studies in
Advanced Mathematics, no. 55, Cambridge University Press, 1996.
[21] D. Bump and J. Hoffstein, On Shimura’s correspondence, Duke Math. J. 55
(1987), no. 3, 661–691.
[22] ˙I. N. Cangül and O. Bizim, Cycloidal normal subgroups of Hecke groups
of finite index, Turkish J. Math. 23 (1999), no. 2, 345–354.
[23] ˙I. N. Cangül and D. Singerman, Normal subgroups of Hecke groups and
regular maps, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 1, 59–74.
[24] Y. Chuman, Generators and Relations of Γ0(N), J. Math. Kyoto Univ.
(1973), 381–390.
[25] H. Cohn, Advanced Number Theory, Dover, 1980.
170

[26] M. Conder and P. Dobcsányi, Normal Subgroups of Low Index in the Modular
Group and other Hecke Groups, Research Report Series 496, Dept. of
Math., Univ. of Auckland, 2003.
[27] D. Davenport, Multiplicative Number Theory, 2 ed., Graduate Texts in
Mathematics, no. 74, Springer-Verlag, 1980.
[28] W. Duke, Hyperbolic distribution problems and half-integral weight
Maass forms, Invent. Math. 92 (1988), 73–90.
[29] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental
Functions. Vols. I, II, McGraw-Hill, 1953.
[30] D.W. Farmer and S. Lemurell, Deformations of Maass forms,
http://xxx.lanl.gov/abs/math.NT/0302214.
[31] L. R. Ford, Automorphic Functions, 2nd ed., Chelsea Publishing, 1972.
[32] W. Gautschi, Computational aspects of three-term recurrence relations,
SIAM Review 9 (1967), no. 1, 24–82.
[33] A. Gil, J. Segura, and N. M. Temme, Evaluation of the modified Bessel function
of the third kind of imaginary orders, J. Comput. Phys. 175 (2002),
no. 2, 398–411.
[34] R. C. Gunning, Lectures on Modular Forms, Princeton University Press,
1962.
[35] M. C. Gutzwiller, Chaos in classical and quantum mechanics, Interdisciplinary
Applied Mathematics, vol. 1, Springer-Verlag, New York, 1990.
[36] E. Hecke, Matematische Werke, Vandhoeck & Ruprecht, Göttingen, 1970.
[37] D. A. Hejhal, Some Dirichlet series with coefficients related to periods
of automorphic eigenforms. I,II, Proc. Japan Acad. Ser. A, 58 (1982), pp.
413–417; 59 (1983), pp. 335-338.
[38] , The Selberg Trace Formula for PSL(2,R), Vol.1, Lecture Notes
in Mathematics, vol. 548, Springer-Verlag, 1976.
[39] , The Selberg Trace Formula for PSL(2,R), Vol.2, Lecture Notes
in Mathematics, vol. 1001, Springer-Verlag, 1983.
[40] , A continuity method for spectral theory on Fuchsian groups,
Modular forms (R.A. Rankin, ed.), Ellis-Horwood, Chichester, 1984, pp. 107–
140.
171

[41] , Regular b-groups, degenerating Riemann surfaces, and spectral
theory, Mem. Amer. Math. Soc. 88 (1990), no. 437, iv+138.
[42] , Eigenvalues of the Laplacian for Hecke triangle groups, Mem.
Amer. Math. Soc. 97 (1992), no. 469, vi+165.
[43] , On eigenfunctions of the Laplacian for Hecke triangle groups,
Emerging applications of number theory (D. Hejhal, J. Friedman, et al., eds.),
IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 291–315.
[44] , On the Calculation of Maass Cusp Forms, Proceedings of the
"International School on Mathematical Aspects of Quantum Chaos II", Lecture
Notes in Physics, Springer, 2004, to appear.
[45] D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for
PSL(2,Z), Math. Comp. 61 (1993), no. 203, 245–267, S11–S16.
[46] D. A. Hejhal and A. Strömbergsson, On quantum chaos and Maass waveforms
of CM-type, Foundations of Physics 31 (2001), 519–533.
[47] T. Hsu, Identifying congruence subgroups of the modular group, Proc.
Amer. Math. Soc. 124 (1996), no. 5, 1351–1359.
[48] M. N. Huxley, Scattering matrices for congruence subgroups, Modular
forms (R. Rankin, ed.), Ellis Horwood, Chichester, 1984, pp. 141–156.
[49] F. M. Ives, Permutation enumeration: four new permutation algorithms,
Commun. ACM 19 (1976), no. 2, 68–72.
[50] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms,
Revista Matemaática Iberoamericana, 1995.
[51] , Topics in Classical Automorphic Forms, American Mathematical
Society, 1997.
[52] S. Katok, Fuchsian Groups, The University of Chicago Press, 1992.
[53] S. Katok and P. Sarnak, Heegner points, cycles and Maass forms, Israel
Journal of Mathematics 84 (1993), 193–227.
[54] K. Khuri-Makdisi, On the Fourier coefficients of nonholomorphic Hilbert
modular forms of half-integral weight, Duke Math. J. 64 (1996), 399–452.
[55] M.I. Knopp, Modular Functions in Analytic Number Theory, Markham
Publishing Company, 1970.
[56] W. Kohnen, Modular forms of half-integral weight on Γ0(4), Math. Ann.
248 (1980), no. 3, 249–266.
172

[57] , Newforms of half-integral weight, J. Reine Angew. Math. 333
(1982), 32–72.
[58] H. Kojima, Shimura correspondence of Maass wave forms with half integral
weight, Acta Arithmetica 69 (1995), no. 4, 367–385.
[59] , On the fourier coefficients of Maass wave forms of half integral
weight over an imaginary quadratic field, J. reine angew. Math. 526 (2000),
155–179.
[60] S.L. Krushkal, B.N. Apanasov, and N.A. Gusevskii, Kleinian Groups and
Uniformization in Examples and Problems, Translations of Mathematical
Monographs, AMS, 1986.
[61] S. Lang, Algebraic Number Theory, second ed., Graduate Texts in Mathematics,
vol. 110, Springer-Verlag, New York, 1994.
[62] H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen
und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen,
Math. Annalen 121 (1949), 141–183.
[63] , Lectures on Modular Functions of One Complex Variable, second
ed., Tata Institute of Fundamental Research Lectures on Mathematics and
Physics, vol. 29, Tata Institute of Fundamental Research, Bombay, 1983.
[64] W. Magnus and R. Oberhettinger, F. Soni, Formulas and Theorems for
the Special Functions of Mathematical Physics, Third enlarged edition,
Springer-Verlag, 1966.
[65] B. Maskit, Kleinian Groups, Springer-Verlag, 1988.
[66] M. H. Millington, On cycloidal subgroups of the modular group, Proc.
London Math. Soc. (3) 19 (1969), 164–176.
[67] , Subgroups of the classical modular group, J. London Math. Soc.
(2) 1 (1969), 351–357.
[68] T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of
Math. (2) 94 (1971), 174–189.
[69] , Modular Forms, Springer-Verlag, 1997.
[70] T. Mühlenbruch, Systems of Automorphic Forms and Period Functions,
Ph.D. thesis, Mathematisch Instituut, Universitet Utrecht, 2003.
[71] K. E. Muller, Computing the confluent hypergeometric function,
M(a,b,x), Numer. Math. 90 (2001), no. 1, 179–196.
173

[72] S. Niwa, Modular forms of half integral weight and the integral of certain
theta-functions, Nagoya Math. J. 56 (1975), 147–161.
[73] F. W. J. Olver, Error analysis of Miller’s recurrence algorithm, Math.
Comp. 18 (1964), 65–74.
[74] , Error bounds for linear recurrence relations, Math. Comp. 50
(1988), no. 182, 481–499.
[75] A. Orihara, On the Eisenstein series for the principal congruence subgroups,
Nagoya Math. J. 34 (1969), 129–142.
[76] A. L. Orive, Some Presentations for Γ0(N), Conformal Geometry and Dynamics
6 (2002), 33–60.
[77] H. Petersson, Zur analytischen Theorie der Grenzkreisgruppen. I. Grenzkreisgruppen
und Riemannsche Flächen; Theorie der Faktoren- und
Multiplikatorsysteme komplexer Dimension, Math. Ann. 115 (1937), 23–
67.
[78] , Über einen einfachen Typus von Untergruppen der Modulgruppe,
Arch. Math. 4 (1953), 308–315.
[79] , Über die Konstruktion zykloider Kongruenzgruppen in der rationalen
Modulgruppe, J. Reine Angew. Math. 250 (1971), 182–212.
[80] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of
PSL(2,R), Invent. Math. 80 (1985), 339–364.
[81] , The Weyl Theorem and the Deformation of Discrete Groups,
Comm. on Pure and Appl. Math. 38 (1985), 853–866.
[82] H. Poincaré, Science and hypothesis, Dover Publications Inc., New York,
1952.
[83] H. Rademacher, Über die Erzeugenden von Kongruenzuntergruppe der
Modulgruppe, Abhandlungen Hamburg 7 (1929), 134–148.
[84] R. A. Rankin, Modular Forms and Functions, Cambridge University Press,
1976.
[85] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag,
1994.
[86] M. S. Risager, Asymptotic densities of Maass newforms, J. Number Theory
109 (2004), no. 1, 96–119.
174

[87] P. Sarnak, Additive number theory and Maass forms, Number theory
(New York, 1982), Lecture Notes in Math., vol. 1052, Springer, Berlin, 1984,
pp. 286–309.
[88] A. Selberg, Collected papers, vol. 1, Springer-Verlag, 1989.
[89] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms,
Princeton Univ Press, 1971.
[90] , On modular forms of half integral weight, Annals of Math. 97
(1973), 440–481.
[91] , On the Fourier coefficients of Hilbert modular forms of halfintegral
weight, Duke Math. J. 71 (1993), 501–557.
[92] T. Shintani, On construction of holomorphic cusp forms of half integral
weight, Nagoya Math. J. 58 (1975), 83–126.
[93] A. Strömbergsson, The Selberg Trace Formula for SL(2,R) and arbitrary
real weight, unpublished manuscript.
[94] A. Strömbergsson, Studies in the analytical and spectral theory of automorphic
forms, Ph.D. thesis, Uppsala University, Dept. of Math., 2001.
[95] A. Strömbergsson and B. Selander, Sextic coverings of genus two which are
branched at three points, Preprint, 2002.
[96] N. M. Temme, The numerical computation of the confluent hypergeometric
function U(a, b, z), Numer. Math. 41 (1983), no. 1, 63–82.
[97] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications,
vol. I, Springer-Verlag, New York, 1985.
[98] H. Then, Maass cusp forms for large eigenvalues, Math. Comp. 74 (2005),
no. 249, 363–381.
[99] F. G. Tricomi, Fonctions hypergéométriques confluentes, Gauthier-Villars,
1960.
[100] A. B. Venkov, A remark on the discrete spectrum of the automorphic
Laplacian for a generalized cycloidal subgroup of a general Fuchsian
group, J. Soviet Math. 52 (1990), no. 3, 3016–3021.
[101] J. Von Neumann and E. Wigner, Über das Verhalten von Eigenwerten bei
adiabatischen Processen, Phys. Zeit. 30 (1929), 467–470.
[102] J. Wimp, On the Computation of Tricomi’s Ψ Function, Computing 13
(1974), 195–203.
175

[103] K. Wohlfahrt, Über Operatoren Heckescher Art bei Modulformen reeller
Dimension, Math. Nachr. 16 (1957), 233–256.
[104] , An extension of F. Klein’s level concept, Illinois J. Math. 8 (1964),
529–535.
[105] P. G. Zograf, The spectrum of automorphic Laplacians in spaces of
parabolic functions, Dokl. Akad. Nauk SSSR 269 (1983), no. 4, 802–805.
[106] , Small eigenvalues of automorphic Laplacians in spaces of cusp
forms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134
(1984), 157–168, Automorphic functions and number theory, II.
176