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.