Modular Forms

A classical modular form of weight k is a holomorphic function f(z) on the upper half plane in the complex numbers which satisfies

f((az+b)/(cz+d))=(cz+d)^k f(z)

for [a,b;c,d] in a subgroup G of finite index in the group of all two by two integer matrices of determinant 1.

When G contains the subgroup of all matrices congruent to the identity modulo N we say f(z) has level N.

The modular forms of level N and weight k form a finite dimensional vector space.