Let denote the complex upper half plane. acts transitively on by linear fractional transformationsNow 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:
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 pA 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.