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.

## Definition

Let $\mathbb {M} _{k}$ be the space of entire modular forms of weight $k$ and $\mathbb {S} _{k}$ the space of cusp forms.

The mapping $\langle \cdot ,\cdot \rangle :\mathbb {M} _{k}\times \mathbb {S} _{k}\rightarrow \mathbb {C}$ ,

$\langle f,g\rangle :=\int _{\mathrm {F} }f(\tau ){\overline {g(\tau )}}(\operatorname {Im} \tau )^{k}d\nu (\tau )$ is called Petersson inner product, where

$\mathrm {F} =\left\{\tau \in \mathrm {H} :\left|\operatorname {Re} \tau \right|\leq {\frac {1}{2}},\left|\tau \right|\geq 1\right\}$ is a fundamental region of the modular group $\Gamma$ and for $\tau =x+iy$ $d\nu (\tau )=y^{-2}dxdy$ is the hyperbolic volume form.

## Properties

The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.

For the Hecke operators $T_{n}$ , and for forms $f,g$ of level $\Gamma _{0}$ , we have:

$\langle T_{n}f,g\rangle =\langle f,T_{n}g\rangle$ This can be used to show that the space of cusp forms of level $\Gamma _{0}$ has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these forms are all real.