4 1. BASIC FACTS

Remark 1.13. For simplicity we shall refer to a holomorphic modular form as

a modular form. Throughout we shall continue to use the terminology meromor-

phic modular form, weakly holomorphic modular form, and modular function as in

Definitions 1.8 and 1.12.

One of the main subjects of this monograph is the study of the Fourier expansion

of meromorphic modular forms. Since

1 1

0 1

is in each congruence subgroup

considered here, we have the following definition.

Definition 1.14. If f(z) is a meromorphic modular form on a congruence

subgroup Γ, then its Fourier expansion at infinity is the expansion of the form

f(z) =

∞

n≥n0

a(n)qn,

where q :=

e2πiz

.

Throughout we let

(1.4) q :=

e2πiz

.

By Definitions 1.8 and 1.12, it follows that meromorphic (resp. holomorphic

and cusp) modular forms of weight k on a congruence subgroup Γ naturally form

C-vector spaces. We denote the complex vector space of modular forms (resp. cusp

forms) of weight k with respect to Γ1(N) by

(1.5) Mk(Γ1(N)) (resp. Sk(Γ1(N)).

For simplicity, we define

Mk := Mk(Γ1(1)),

Sk := Sk(Γ1(1)).

(1.6)

Of particular interest are certain modular forms in Mk(Γ1(N)) with nice modular

transformation properties with respect to Γ0(N).

Definition 1.15. If χ is a Dirichlet character modulo N, then we say that a

form f(z) ∈ Mk(Γ1(N)) (resp. Sk(Γ1(N)) ) has Nebentypus character χ if

f

az + b

cz + d

= χ(d)(cz +

d)kf(z)

for all z ∈ H and all

a b

c d

∈ Γ0(N). The space of such modular forms (resp.

cusp forms) is denoted by Mk(Γ0(N), χ) (resp. Sk(Γ0(N), χ)).

If χ = χ0 is trivial, then we denote Mk(Γ0(N), χ0) (resp. Sk(Γ0(N), χ0)) by

Mk(Γ0(N)) (resp. Sk(Γ0(N)).

Remark 1.16. If χ is a Dirichlet character modulo N, then we assume that

χ(n) := 0 for every integer n with gcd(n, N) = 1. Furthermore, throughout we let

χ0 denote the trivial character.

Remark 1.17. Since

−1 0

0 −1

∈ Γ0(N), if χ(−1) =

(−1)k,

then there are

no nonzero modular forms in Mk(Γ0(N), χ).