Expansions at Cusps and Petersson Products in Pari/GP

TL;DR

This paper presents methods for computing Fourier expansions at all cusps of modular forms and for calculating Petersson products, implemented in the Pari/GP software package.

Contribution

It introduces new algorithms and explicit formulas for Fourier expansions and Petersson products, integrated into Pari/GP for broader accessibility.

Findings

Efficient computation of Fourier expansions at all cusps.

Explicit formulas for Eisenstein series expansions.

Implementation of methods in Pari/GP package.

Abstract

We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.