Petersson inner product

Summary

The Petersson inner product is an inner product on the space $S_k$ of cusp forms of weight $k$. It integrates $f \overline{g}\, y^k$ against the hyperbolic area over a fundamental domain of the modular group. The Hecke operators are self-adjoint for it, so $S_k$ has a basis of common eigenforms. The Eisenstein series are orthogonal to the cusp forms in a suitable sense.