Petersson inner product

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. Let be the space of entire modular forms of weight and the space of cusp forms. The mapping , is called Petersson inner product, where is a fundamental region of the modular group and for is the hyperbolic volume form. The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form. For the Hecke operators , and for forms of level , 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.