Eta Products and Models for Modular Curves

Theorem 1: Suppose r, d, M, and l are positive divisors of N, such
that Ml ∣ ∣ N (so πl : X 0 (N) → X 0 (M)) and lcm(d, r) = N. Suppose ζ is
a primitive r th root of unity. Suppose f is any weight k modular form
for Γ 0 (M).
Let g = gcd(d, N/l), and find a, b ∈ Z s.t. ad + b(N/l) = g.
If
then
f (C ∗ p/(ζ bNg/d q lg2 /d ), 〈(ζq) lg/d 〉[M]) = f (q) ( )
dz ⊗k
z ,
π ∗ l f (C ∗ p/q d , 〈ζq〉) = f (q)
(
lg
d
) k ( dz
) ⊗k
z .