18 1. BASIC FACTS

where

χ12(n) :=

12

n

=

1

if n ≡ 1, 11 (mod 12),

−1 if n ≡ 5, 7 (mod 12),

0 otherwise.

Definition 1.63. Any function f(z) of the form

f(z) =

δ|N

η(δz)rδ

,

where N ≥ 1 and each rδ is an integer, is known as an eta-quotient. If each rδ ≥ 0,

then f(z) is known as an eta-product.

Using Theorem 1.61, it is not diﬃcult to deduce the modular transformation

properties of eta-quotients. The following general result of Gordon, Hughes, and

Newman [GoH, N1, N3] comes in handy when working with eta-quotients and

eta-products.

Theorem 1.64. If f(z) =

δ|N

η(δz)rδ

is an eta-quotient with k =

1

2

∑

δ|N

rδ ∈

Z, with the additional properties that

δ|N

δrδ ≡ 0 (mod 24)

and

δ|N

N

δ

rδ ≡ 0 (mod 24),

then f(z) satisfies

f

az + b

cz + d

= χ(d)(cz +

d)kf(z)

for every

a b

c d

∈ Γ0(N). Here the character χ is defined by χ(d) :=

((−1)ks)

d

,

where s :=

δ|N

δrδ

.

Suppose that k is a positive integer and that f(z) is an eta-quotient satisfying

the conditions of Theorem 1.64. If f(z) is holomorphic (resp. vanishes) at all of

the cusps of Γ0(N), then f(z) ∈ Mk(Γ0(N), χ) (resp. Sk(Γ0(N), χ)). Since η(z)

is analytic and never vanishes on H, it suﬃces to check that the orders at the

cusps are nonnegative (resp. positive). The following theorem (see, for example

[Bi, Lig, Ma]) is the necessary criterion for determining orders of an eta-quotient

at cusps in terms of the usual local variables.

Theorem 1.65. Let c, d and N be positive integers with d | N and gcd(c, d) = 1.

If f(z) is an eta-quotient satisfying the conditions of Theorem 1.64 for N, then the

order of vanishing of f(z) at the cusp

c

d

is

N

24

δ|N

gcd(d,

δ)2rδ

gcd(d,

N

d

)dδ

.