Let denote the complex upper half plane. acts transitively on by linear fractional transformations

Now we consider functions . The notation is used to denote the function whose value at is . The value of at is denoted by so that

We define modular forms and cusp forms for a congruence subgroup of level N. Let denote . Let . A function is said to be a modular form of weight k for if it satisfies the following conditions:
(i)
f is holomorphic on ,
(ii)
for all ,
(iii)
f is holomorphic at the cusps for ; i.e. has the form .

We let denote the set of such modular forms. It turns out that is a finite dimensional -vector space. If for all we say f is a cusp form. We let denote the subspace of cusp forms.

Now we define what it means for a function to be a modular form of weight k and character for the group when p is prime. Let be a Dirichlet character modulo p

A function is said to be a modular form of weight k and character for if it satisfies conditions (i)--(iii) above except (ii) is replaced by

In this case, condition (iii) amounts to the following two conditions:

We call the Fourier series in (1) the q-expansion of f at , and we call (2) the q-expansion at 0. We let denote the space of such modular forms. If above then f is a cusp form. We let denote the subspace of cusp forms.