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
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
where q :=
Throughout we let
(1.4) q :=
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)).
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
az + b
cz + d
= χ(d)(cz +
for all z ∈ H and all
∈ 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
∈ Γ0(N), if χ(−1) =
then there are
no nonzero modular forms in Mk(Γ0(N), χ).