Cusp form

Summary

A cusp form is a modular form that vanishes at every cusp. For the full modular group, this means that the constant term of its $q$-expansion is $0$. The cusp forms of weight $k$ form a subspace $S_k$ of $M_k$, on which the Petersson inner product is defined. The coefficients of cusp forms grow more slowly than those of Eisenstein series, and they carry deep arithmetic information.