Numerical computation of Petersson inner products and $q$-expansions

Abstract

In this paper we discuss the problem of numerically computing Petersson inner products of modular forms, given their $q$-expansion at $\infty$. A formula of Nelson reduces this to obtaining $q$-expansions at all cusps, and we describe two algorithms based on linear interpolation for numerically obtaining such expansions. We apply our methods to numerically verify constants arising in an explicit version of Ichino's triple-product formula relating $\langle fg,h\rangle$ to the central value of $L(f\times g\times \bar{h},s)$, for three modular forms $f,g,h$ of compatible weights and characters.