Shimura lifts of half-integral weight modular forms - Department of ...

SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS
ARISING FROM THETA FUNCTIONS
DAVID HANSEN AND YUSRA NAQVI
Abstract. In 1973, Shimura [8] introduced a family of correspondences between modular
forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished
work, Selberg explicitly computed a simple case of this correspondence pertaining
to those half-integral weight forms which are products of Jacobi’s theta function and level
one Hecke eigenforms. Cipra [1] generalized Selberg’s work to cover the Shimura lifts
where the Jacobi theta function may be replaced by theta functions attached to Dirichlet
characters of prime power modulus, and where the level one Hecke eigenforms are replaced
by more generic newforms. Here we generalize Cipra’s results further to cover theta functions
of arbitrary Dirichlet characters multiplied by more general Hecke eigenforms, and
we use these explicit formulas to compute optimal levels for these lifts without appealing
to Shimura’s deeper arguments.
1. Introduction and statement of results
Let SL2(Z) denote the set of all 2-by-2 matrices with integer entries and determinant
1, and let k be a positive integer. We say that f(z) is a modular form of weight k on the
congruence subgroup Γ0(N) with multiplier ψ if f(z) is a holomorphic function on the upper
half of the complex plane which satisfies f(γz) = (cz + d) kψ(d)f(z) for all γ =
a b
c d ∈
SL2(Z) with c ≡ 0 (mod N). Let Mk(N, ψ) denote the finite-dimensional vector space of
modular forms of weight k on Γ0(N) with multiplier ψ, where ψ is a Dirichlet character of
modulus N. A modular form is called a cusp form if it vanishes at all rational points and
at infinity. We let Sk(N, ψ) denote the subspace of Mk(N, ψ) consisting only of cusp forms.
For k ≥ 2 a positive even integer, we define the Eisenstein series of weight k by
(1.1) Ek(z) := 1 − 2k
Bk
n=1
∞
σk−1(n)q n ,
where Bn in the nth Bernoulli number and q := e 2πinz . These functions represent the
simplest modular forms of weight k, and they lie in Mk(1, 1). More general Eisenstein
series can be defined as follows. Let χ1 (mod a1) and χ2 (mod a2) be primitive Dirichlet
characters of conductors a1, a2, not both trivial, and set a = a1a2. The character χ = χ1χ2
has modulus a. If k is positive integer with χ(−1) = (−1) k , then set
(1.2) Ek(χ1, χ2; z) := C(k, χ1, χ2) +
∞
n=1
1
d|n
χ1(n/d)χ2(d)d k−1
q n ,

2 DAVID HANSEN AND YUSRA NAQVI
where C(k, χ1, χ2) is zero unless a1 = 1, in which case C(k, χ1, χ2) = 1
2L(1 − k, χ),
where L(s, χ) = ∞ n=1 χ(n)n−s is the Dirichlet L-function of the character χ. We have
Ek(χ1, χ2; z) ∈ Mk(a, χ); see Chapter 4 of [4].
In a classic paper [8], Shimura invented the modern theory of modular forms of halfintegral
weight. Briefly, let N, k be positive integers with ψ a Dirichlet character of modulus
4N. We say that f is a modular form of weight k + 1/2 with multiplier ψ if
c
2k+1 (1.3) f(γz) = ψ(d) ɛ
d
−2k−1
d (cz + d) k+1/2 f(z)
for all γ ∈ Γ0(4N), where ɛd is 1 or i for odd d according to whether d ≡ 1 (mod 4) or
d ≡ 3 (mod 4), respectively, and
c
d is Shimura’s extension of the Jacobi symbol. As above,
Mk+1/2(N, ψ) denotes the finite-dimensional vector space of weight k + 1/2 modular forms,
and Sk+1/2(N, ψ) denotes its subspace of cusp forms.
Define theta functions
(1.4) θ(χ; z) :=
χ(n)n ν q n2
∈ M1/2+ν(4r 2 , χχ ν 4)
n∈Z
for χ a Dirichlet character of modulus r, where ν = 0, 1 is chosen such that χ(−1) = (−1) ν .
These functions are the simplest examples of modular forms of half-integral weight, and for
k = 1/2, the space is spanned by them (c.f. [7]). For a good introduction to this material,
see [6].
Shimura also established a family of nontrivial maps between modular forms of halfintegral
weight and modular forms of even integer weight. These maps, known as the
Shimura lifts, can be stated as follows.
Theorem (Shimura). Let t be a positive squarefree integer, and suppose that f(z) =
∞
n=1 b(n)qn ∈ Sk+1/2(4N, ψ), where k is a positive integer. If numbers A(n) are defined by
∞
(1.5)
A(n)n −s := L(s − k + 1, ψχ k ∞
4χt) b(tn 2 )n −s ,
n=1
where χt =
t
• is the usual Kronecker character modulo t, then St(f)(z) := ∞ n=1 A(n)qn ∈
M2k(2N, ψ2 ). Moreover, if k > 1, then St(f)(z) is a cusp form.
Shimura lifts play an important role in several areas of modern number theory, including
Tunnell’s famous work [9] on the ancient congruent number problem, and recent work by
Ono [5] on congruences for the partition function. Moreover, in these particular applications,
the relevant half-integral weight forms can be written as products of integer weight forms
and theta functions. In light of these facts, it is desirable to have explicit formulas for the
Shimura lifts in these cases.
It turns out that much earlier, in unpublished work, Selberg worked out such an explicit
formula. Briefly, for certain modular forms f(z) ∈ Mk(1, 1), Selberg found that
f(4z)θ(1; z) ∈ M k+1/2(4, 1) lifts to f(z) 2 − 2 k−1 f(2z) 2 ∈ M2k(2, 1). Later on, Cipra [1]
generalized Selberg’s work by proving the following result.
n=1

SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 3
Theorem (Cipra). If f(z) ∈ Sk(N, ψ) is a newform, and θ(χr; z) is the theta function
of an even Dirichlet character of prime power modulus r = p m , then if we define g(z) :=
f(z)f(p µ z), the Shimura lift S1 of f(4p µ z)θ(χr; z) is
(1.6) gχr(z) − 2 k−1 χr(2)ψ(2)gχr(2z),
where µ is any integer with µ ≥ m.
Cipra also proves a similar statement for theta functions with odd characters. However,
Cipra’s class of eligible forms f(z) is limited to newforms, and his use of theta functions with
characters to prime power moduli is a highly restrictive condition. We prove the following
two theorems, generalizing these results.
Theorem 1.1. Let χr be an even Dirichlet character modulo r, and write χr = χ p α 1
1 χ p α 2
2 ...χ p α j
j
as the factorization of χr into Dirichlet characters modulo prime powers p α1
1
, pα2
2
, ..., pαj
j
with p α1
1 pα2
2 ...pαj
j = r. Let f(z) ∈ Mk(N, ψ) be a Hecke eigenform, and set F (z) :=
θ(χr; z)f(4rz) ∈ M k+1/2(4N ′ r 2 , ψχrχ k 4 ) with N ′ = N/ gcd(N, r). If
(1.7) g(z) :=
where χd =
p αj j ||d χ p αj j
, then we have
d|r
gcd(d,r/d)=1
f(dz)f(rz/d)χd(−1),
(1.8) S1(F )(z) = gχr(z) − 2 k−1 χr(2)ψ(2)gχr(2z) ∈ M2k(2N ′ r 2 , ψ 2 χ 2 r).
Here gχ is the χ-twist of g.
For the case of odd characters, the theorem is slightly different, due to the fact that the
relevant theta functions now have weight 3/2.
Theorem 1.2. Let χr be an odd Dirichlet character modulo r, and write χr = χ p α 1
1 χ p α 2
2 ...χ p α j
j
as the factorization of χr into Dirichlet characters modulo prime powers p α1
1
with p α1
1 pα2
2
, pα2
2
, · · · , pαj
j
· · · pαj
j = r. If F (z) := θ(χr; z)f(4rz) ∈ M k+3/2(4N ′ r 2 , ψχrχ k+1
4 ), where
f(z) ∈ Mk(N, ψ) is a Hecke eigenform, and
(1.9) g(z) := 1
πi
where χd =
p αj j ||d χ p αj j
, then we have
d|r
gcd(d,r/d)=1
df ′ (dz)f(rz/d)χd(−1),
(1.10) S1(F )(z) = gχr(z) − 2 k χr(2)ψ(2)gχr(2z) ∈ M2k+2(2N ′ r 2 , ψ 2 χ 2 r),
where gχ is the χ-twist of g.
The proofs of our theorems, like those of Selberg and Cipra, are entirely combinatorial,
using only elementary properties of Dirichlet series and a multiplicativity relation for the
coefficients of our starting form f(z). This multiplicativity is conditional on f(z) being
a Hecke eigenform. However, since any given modular form can be written as a linear
combination of eigenforms, our theorems can be applied to more general products of modular

4 DAVID HANSEN AND YUSRA NAQVI
forms and theta functions by the linearity of the Shimura lift. Furthermore, we compute the
levels of the lifts in Theorems 1.1 and 1.2 directly, without appealing to any of Shimura’s
results. In fact, our theorems are completely independent of Shimura’s work.
In Section 2, we define and explain the notion of a Hecke eigenform and the associated
multiplicativity relations for its coefficients. In Section 3, we present proofs of Theorems
1.1 and 1.2, and we discuss a method of determining the cuspidality of the lifts given by
these theorems. We also show how to obtain the optimal level for the lifted forms. Section
4 contains a discussion of examples and applications.
2. Multiplicativity Properties of Modular Form Coefficients
Let f(z) = ∞
n=0 a(n)qn ∈ Mk(N, ψ). The action of the nth Hecke operator T ψ n of weight
k on f(z) is given by
(2.1) f(z) | T ψ n =
∞
m=0
d|(m,n)
ψ(d)d k−1 a(mn/d 2
) q m .
It is known, by work of Hecke, that these operators are linear endomorphisms on Mk(N, ψ).
They also map cusp forms to cusp forms. Furthermore, there exist modular forms f(z) ∈
Mk(N, ψ) which are simultaneous eigenfunctions of all the Hecke operators; in other words,
they satisfy
(2.2) f(z) | T ψ n = λ(n)f(z)
for all positive integers n, where the λ(n) are complex numbers. If f(z) satisfies these
conditions, we generally refer to it as a Hecke eigenform. In this case, by combining (2.1)
and (2.2), we easily get
(2.3) λ(n)a(m) =
d|(m,n)
ψ(d)d k−1 a(mn/d 2 ).
If a(1) = 1, then this reveals that in fact λ(n) = a(n), and we can then reformulate (2.3)
as follows.
Proposition 2.1. If f(z) = ∞
n=0 a(n)qn ∈ Mk(N, χ) is a simultaneous eigenfunction of
all the Hecke operators T ψ n with a(1) = 1, then for any positive integers m, n, we have
a(m)a(n) =
d|(m,n)
ψ(d)d k−1 a(mn/d 2 ).

SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 5
For the Eisenstein series defined in the introduction (see 1.2), we can prove this directly
as follows:
d|(m,n)
=
d|(m,n)
=
d|(m,n)
d 2 δ|mn
=
d|(m,n)
d 2 δ|mn
χ(d)d k−1 a(mn/d 2 )
k−1
χ(d)d
δ|mn/d 2
χ1(mn/(d 2 δ))χ2(δ)δ k−1
χ1(d)χ2(d)χ1(mn/(d 2 δ))χ2(δ)d k−1 δ k−1
χ1(mn/(dδ))χ2(dδ)(dδ) k−1
=
χ1(mn/(dδ))χ2(dδ)(dδ) k−1
d|m
δ|n
= a(m)a(n).
Furthermore, we have an “inverse” of Proposition 2.2, which we shall refer to as Selberg
inversion.
Proposition 2.2. If f(z) = ∞ n=0 a(n)qn ∈ Mk(N, χ) is a Hecke eigenform with a(1) = 1,
then we have
a(mn) =
µ(d)χ(d)d k−1 a(m/d)a(n/d),
for any positive integers m, n.
Proof. We have that
d|(m,n)
=
d|(m,n)
=
dδ|(m,n)
=
D|(m,n)
= a(mn).
d|(m,n)
µ(d)χ(d)d k−1 a(m/d)a(n/d)
µ(d)χ(d)d k−1
δ|(m/d,n/d)
µ(d)χ(dδ)(dδ) k−1 a(mn/(dδ) 2 )
d|D
µ(d) χ(D)D k−1 a(mn/D 2 )
χ(δ)δ k−1 a(mn/(dδ) 2 )

6 DAVID HANSEN AND YUSRA NAQVI
3. Proofs of Theorems 1.1 and 1.2
We begin by presenting the proof of the formula for the lift in Theorem 1.1. From the
definition of F (z), we have F (z) = ∞ n=0 b(n)qn with
(3.1) b(n) =
n − m2
χr(m)a ,
4r
m∈Z
where f(z) = ∞ n=0 a(n)qn is as in the statement of Theorem 1.1. As above, the Shimura
lift is given by
∞
(3.2) S1(F ) = A(n)q n
with the coefficients A(n) defined by
(3.3)
∞
n=1
n=1
A(n)n −s = L(s − k + 1, χrψχ 2k
4 )
We also need the coefficients defined by
∞
(3.4)
cd(n)q n := f(dz)f(rz/d) =
n=0
n=0 m∈Z
∞
b(n 2 )n −s .
n=1
∞
n − dm
a(m)a q
r/d
n .
Throughout the proof, we use the convention that a modular form coefficient is zero if its
argument is negative or not integral. From (3.1) it is easy to see that
(3.5) b(n 2 ) =
(n − m)(n + m)
χr(m)a
.
4r
m∈Z
This is a finite sum with non-zero coefficients whenever (n − m)(n + m)/(4r) ∈ N. Note
that n + m and n − m must both be even for n and m to be integers with 4|(n 2 − m 2 ). Let
gcd((n − m)/2, r) = d. Thus, m ≡ n (mod 2d) and m ≡ −n (mod 2r/d). Now suppose
gcd(d, r/d) = d ′ > 1. This implies that m ≡ n ≡ −n ≡ 0 (mod 2d ′ ), so d| gcd(m, r) and
so χr(m) = 0. Therefore, we only consider the cases in which gcd(d, r/d) = 1. We have
m = n + 2dm ′ for some m ′ ∈ Z, so n − m = −2dm ′ and n + m = 2n + 2dm ′ . Thus,
(3.6)
(n − m)(n + m)
4r
= −m′ (n + dm ′ )
.
r/d
Also, since m ≡ n (mod d) and m ≡ −n (mod r/d), we have that
= χd(m)χ r/d(m)
= χ r/d(−1)χd(n)χ r/d(n)
χr(m)
= χd(n)χ r/d(−n)
= χ r/d(−1)χr(n),

SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 7
where the characters are as defined in the statement of Theorem 1.1. Since χr(−1) =
χ r/d(−1)χd(−1) = 1, we have χd(−1) = χ r/d(−1) = ±1. Thus, by changing the variable
m ′ to −m, (3.5) becomes
(3.7) b(n 2 ) = χr(n)
d|r
gcd(d,r/d)=1
We now apply Proposition 2.2 to get
b(n 2 ) = χr(n)
χd(−1)
= χr(n)
= χr(n)
d|r
gcd(d,r/d)=1
d|r
gcd(d,r/d)=1
d|r
gcd(d,r/d)=1
χd(−1)
χd(−1)
m(n − dm)
a
.
r/d
m∈Z δ|(m,n)
δ|n
m∈Z
µ(δ)ψ(δ)δ k−1
m
n − dm
a a
δr/d δr/d
k−1
µ(δ)ψ(δ)δ
m∈Z
χd(−1)
µ(δ)ψ(δ)δ k−1 cd(n/δ).
Rewriting these formulas as Dirichlet series immediately gives
(3.8)
∞
b(n 2 )n −s =
∞
χd(−1)
n=1
n=1
d|r
gcd(d,r/d)=1
δ|n
n=1 δ|n
m
a a
r/d
n/δ − dm
r/d
µ(δ)ψ(δ)χr(δ)δ k−1 χr(n/δ)cd(n/δ)n −s ,
and we can easily pull out the reciprocal of a Dirichlet L-function to produce
(3.9)
∞
b(n 2 )n −s 1
=
L(s − k + 1, ψχr)
∞
χd(−1) χr(n)cd(n)n −s .
n=1
d|r
gcd(d,r/d)=1
Multiplying by L(s − k + 1, χrψχ2k 4 ) = L(s − k + 1, χrψχ2 4 ), as in the definition of the
Shimura lift, we have
(3.10)
∞
A(n)n −s = L(s − k + 1, χrψχ2 4 )
L(s − k + 1, χrψ)
∞
χd(−1) χr(n)cd(n)n −s .
d|r
gcd(d,r/d)=1
By an easy consideration of Euler products, the quotient of the L-functions simplifies to
1 − 2 k−1−s χr(2)ψ(2), and rewriting into q-series completes the proof of the identity for the
lift.
The proof of the equation for the lift in Theorem 1.2 follows along the same lines, with
appropriate changes due to the slightly different expression for the theta function. The
congruence condition reasoning following (3.5) does not change, and (3.7) becomes
(3.11) b(n 2 ) = χr(n)
χr/d(−1)
m(n − dm)
a
(n − 2dm).
r/d
d|r
gcd(d,r/d)=1
m∈Z
n=1
n=1

8 DAVID HANSEN AND YUSRA NAQVI
Recall that χr is odd here, so χr(−1) = χ r/d(−1)χd(−1) = −1, and so we have that
χd(−1) = −χ r/d(−1) = ±1. Selberg inversion applies again, and the derivatives of modular
forms appearing in the definition of g(z) arise naturally from the linear form in m and n
appearing in (3.11).
In the odd case, it is not immediately clear that g(z) is in fact a modular form, since it
contains derivatives of modular forms. However, it is in fact easy to prove modularity by
employing the following useful fact.
Proposition 3.1. Let f(z) be a modular form of weight k on some subgroup of SL2(Z).
Then 1 d
2πi dz f(z) = ( ˜ f(z) + kE2(z)f(z))/12, where E2(z) is the Eisenstein series defined in
(1.1) and ˜ f(z) is a modular form of weight k + 2.
Note that E2 is not a modular form (see [6]). Using this proposition, we easily obtain
g(z) = 1
πi
χd(−1)df ′ (dz)f(rz/d)
= 1
2π 2 i 2
= 1
12πi
= 1
12πi
d|r
gcd(d,r/d)=1
d|r
gcd(d,r/d)=1
d|r
gcd(d,r/d)=1
d|r
gcd(d,r/d)=1
χd(−1)f(rz/d) ∂
∂z f(dz)
χd(−1)f(rz/d)( ˜ fd(z) + kE2(z)f(dz))
χd(−1)f(rz/d) ˜ fd(z),
where ˜ fd(z) is a modular form of weight k+2 and level dN. The sum involving E2’s vanishes
due to cancellation in characters, namely χd(−1) = −χ r/d(−1).
To complete the proofs of Theorems 1.1 and 1.2, it suffies to compute the levels of the
relevant Shimura lifts. Because g(z) lies in the space M2k(N ′ r, ψ 2 ), it is easy to see by the
general theory of twists (see [6], Sec. 2.2) that gχr(z) ∈ M2k(N ′ r 3 , χ 2 rψ 2 ). However, we can
in fact show that gχr(z) lies in the space M2k(N ′ r 2 , χ 2 rψ 2 ). To do this, we demonstrate the
invariance of gχr(z) under a complete set of representatives of right cosets of Γ0(N ′ r 3 ) in
Γ0(N ′ r 2 ). By Proposition 2.5 of [2], we have that [Γ0(N ′ r 2 ) : Γ0(N ′ r 3 )] = r, so such a set
of representatives is given by
(3.12) αj :=
1 0
jN ′ r2
1
for j = 0, 1, 2, ..., r − 1. For convenience, we define the slash operator for γ ∈ GL + 2 (Q) by
(3.13) f(z) |k γ := f(γz)(cz + d) −k (det γ) k/2 .
With this notation, we need to show gχr(z) |k αj = gχr(z) for j = 0, 1, 2, ..., r − 1. Using
Proposition 17 in Sec. 3.3 of [3] and defining τ(χr) := r−1
m=0 χr(m)e 2πim/r , we first write

SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 9
gχr(z) as a sum of linear transforms,
where we have set
gχr(z) = r −1 r−1
τ(χr)
v=0
= r −1 r−1
τ(χr)
(3.14) γv :=
It then follows that
gχr(z) | αj = r −1 r−1
τ(χr)
= r −1 r−1
τ(χr)
v=0
= r −1 r−1
τ(χr)
v=0
= r −1 r−1
τ(χr)
= gχr(z).
v=0
v=0
¯χr(v)g(z)
v=0
1 −v/r
.
0 1
¯χr(v)g(z)
k
¯χr(v)g(z)
k
¯χr(v)g(z) k γv
1 −v/r
0 1
¯χr(v)g(z − v/r)
¯χr(v)g(z) | γv,
k γvαj
1 0
jN ′ r 2 1
1 − jvN ′ r −jN ′ v 2
jN ′ r 2 jvN ′ r + 1
1 −v/r
0 1
Note that the first matrix in the fourth line is in Γ0(N ′ r) with d ≡ 1 (mod N ′ ), and so it
has an invariant action on g(z).
Having an explicit form for the lift allows us to check its cuspidality directly, without
using the analytic machinery of Shimura’s theorem. If f(z) is a cusp form, then it is easy to
see that S1(F )(z) must also be a cusp form, since a sum of cusp forms is itself cuspidal. We
now consider, as a simple example, the case in which f(z) ∈ Mk(1, 1) is a Hecke eigenform
that is not a cusp form. Let F (z) = θ(χr; z)f(4rz), and recall that
(3.15) g(z) =
χδ(−1)f(δz)f(rz/δ).
δ|r
gcd(δ,r/δ)=1
Also recall that if 2|r, then we have that S1(F )(z) = gχ(z). If r is odd, then we define
h(z) := g(z) − 2k−1g(2z), noting that in this case, the Shimura lift is hχ(z). We shall
proceed by computing the Fourier expansions of g(z) and h(z) around a complete set of
cusps.
Let gγ(z) denote g(z) |2k γ. For any γ =
a b
c d ∈ SL2(Z), we have
a b
(3.16) f(δz) = δ
c d
−k/2
δ 0 a b
f(z)
.
0 1 c d
k
k

10 DAVID HANSEN AND YUSRA NAQVI
Let δ ′ = gcd(c, δ). We have that there exists an integer y such that (δ/δ ′ )|(cy + d), and so
we get
f(δz)
k
a b
c d
= δ −k/2
f(z)
Inserting this into the definition of g(z) gives
g(z)
2k
a b
=
c d
=
δ|r
gcd(δ,r/δ)=1
δ|r
gcd(δ,r/δ)=1
aδ/δ ′ δ
k
′ (ay + b)
c/δ ′ δ ′ (cy + d)/δ
= (δ/δ ′ ) −k
δ ′2z − δ ′ y
f
χδ(−1)f(δz)f(rz/δ)
δ
2k
.
a b
c d
δ
−k r/δ
−k χδ(−1)
f
(δ, c) (r/δ, c)
δ ′ −y
0 δ/δ ′
δ ′2 z − y ′
δ
f
δ ′′2 z − y ′′
where δ ′ is as before, δ ′′ = (r/δ, c) and y ′ and y ′′ are integers that depend on δ. This
transforms into
(3.17) gγ(z) =
δ|r
gcd(δ,r/δ)=1
r
−k χδ(−1) f
(r, c)
δ ′2 z − y ′
δ
f
δ ′′2 z − y ′′
We now consider
(3.18)
a
g(2z)
2k c
b
= 2
d
−k
2
g(z)
2k 0
0 a
1 c
b
,
d
which gives us that g(2z) |2k
a b
c d = gγ(2z) if c is even or gγ((z − x)/2) if c is odd, where
x is some integer that depends on d. This yields
h(z)
2k
a b
= gγ(z) − 2
c d
k−1 gγ(2z) or
= gγ(z) − 2 −k−1 gγ((z − x)/2).
Thus, in all cases, the constant term of the Fourier expansion is a constant multiple of
( r
(r,c) )−k a(0) 2 χδ(−1), and hence this term vanishes if and only if f is a cusp form or
(3.19)
δ|r
gcd(δ,r/δ)=1
χδ(−1) = 0.
In particular, this sum vanishes if and only if χr decomposes into a product of Dirichlet
characters to prime power moduli which are not all even. Note that by [7], this is equivalent
to θ(χr; z) being a cusp form. This same method can be applied to modular forms of higher
level; however, the computations are more complicated.
δ
.
δ
,

SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 11
4. Examples and Applications
In this section, we present some examples illustrating Theorems 1.1 and 1.2 in which we
calculate the explicit form of the Shimura lift for certain half-integral weight forms.
Example 1. We begin with Theorem 1.1, by defining f(z) := η(z) 5 /η(5z), where
η(z) := q
1/24
n>0
(1 − q n )
is the Dedekind eta function. This function is in the space M2(5, χ5), and in fact we have
f(z) = −5E2(1, χ5; z). We compute the Shimura lift of f(48z)η(24z) = 1
2f(48z)θ(χ12; z).
To utilize Theorem 1.1, we factor χ12 = χ−4χ−3 and 12 = 22 · 3 to obtain
(4.1) g(z) = f(z)f(12z) − f(3z)f(4z).
Because χ12(2) = 0, the second term in (1.7) vanishes and we have
(4.2)
η(48z) 5η(24z)
η(z) 5η(12z) 5
S1
=
η(240z) η(5z)η(60z) − η(3z)5η(4z) 5
= 25q
η(15z)η(20z) χ12
7 +50q 11 +100q 13 +150q 17 +...
Example 2. We now illustrate Theorem 1.2 by computing the lift of ∆(60z) 2 θ(χ−15; z),
where ∆(z) = η(z) 24 is the standard discriminant function. To apply our theorem, we must
write ∆(z) 2 as a linear combination of Hecke eigenforms. The two Hecke eigenforms, say
f1(z) and f2(z), of weight 24 and level 1 have Fourier expansions
(4.3) fi(z) = q + aiq 2 + (195660 − 48ai)q 3 + ...
with a1 = 540 + 12 √ 144169 and a2 = 540 − 12 √ 144169. Hence,
(4.4) ∆(z) 2 = f1(z) − f2(z)
24 √ 144169 .
To apply Theorem 1.2, we factor 15 = 3 · 5 and χ−15 = χ−3χ5 to obtain
(4.5) g(z) = 1
πi
f ′ 1(z)f1(15z) − 3f ′ 1(3z)f1(5z) + 5f ′ 1(5z)f1(3z) − 15f ′ 1(15z)f1(z)
and hence S1(θ(χ−15; z)f1(60z))(z) = gχ−15 (z) + 224gχ−15 (2z). A similar formula holds for
the lift of f2(z).
References
[1] B. A. Cipra. On the Shimura lift, après Selberg. J. Number Theory, 32(1):58–64, 1989.
[2] H. Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American
Mathematical Society, Providence, RI, 1997.
[3] N. Koblitz. Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics.
Springer-Verlag, New York, second edition, 1993.
[4] T. Miyake. Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition,
2006. Translated from the 1976 Japanese original by Yoshitaka Maeda.
[5] K. Ono. Distribution of the partition function modulo m. Ann. of Math. (2), 151(1):293–307, 2000.
[6] K. Ono. The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume
102 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the
Mathematical Sciences, Washington, DC, 2004.

12 DAVID HANSEN AND YUSRA NAQVI
[7] J.-P. Serre and H. M. Stark. Modular forms of weight 1/2. In Modular functions of one variable, VI
(Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pages 27–67. Lecture Notes in Math., Vol. 627.
Springer, Berlin, 1977.
[8] G. Shimura. On modular forms of half integral weight. Ann. of Math. (2), 97:440–481, 1973.
[9] J. B. Tunnell. A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.,
72(2):323–334, 1983.
Department of Mathematics, Brown University, Providence, RI 02912
E-mail address: david hansen@brown.edu
Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081
E-mail address: yusra.naqvi@gmail.com