In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms.
It was introduced by the German mathematician Hans Petersson.
be the space of entire modular forms of weight
, is called Petersson inner product, where is a fundamental region of the modular group
The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.
, we have: This can be used to show that the space of cusp forms of level
has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these forms are all real.