Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

TL;DR

This paper develops elementary methods to explicitly compute Petersson products of cusp form bases, providing bounds for Fourier coefficients without relying on Rankin L-functions, advancing understanding of cusp form structures.

Contribution

It introduces a new elementary approach to compute Petersson products and bounds Fourier coefficients for non-newform cusp forms, avoiding complex L-function techniques.

Findings

Explicit orthogonal basis for cusp forms constructed

Bounds for Fourier coefficients established

Elementary methods successfully applied to Petersson product computation

Abstract

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.