Bounds for coefficients of cusp forms and extremal lattices

TL;DR

This paper establishes explicit bounds on Fourier coefficients of cusp forms of weight k, and applies these bounds to identify the largest weight for which a certain modular form has non-negative coefficients, impacting extremal lattice theory.

Contribution

It provides a new explicit bound on Fourier coefficients of cusp forms based on initial coefficients, and determines the maximal weight with non-negative coefficients for a specific modular form.

Findings

Explicit bounds on Fourier coefficients in terms of initial coefficients.

Identification of the largest weight with non-negative coefficients as 81632.

Applications to the theory of extremal lattices.

Abstract

A cusp form $f(z)$ of weight $k$ for $\SL_{2}(\Z)$ is determined uniquely by its first $\ell := \dim S_{k}$ Fourier coefficients. We derive an explicit bound on the $n$th coefficient of $f$ in terms of its first $\ell$ coefficients. We use this result to study the non-negativity of the coefficients of the unique modular form of weight $k$ with Fourier expansion \[F_{k,0}(z) = 1 + O(q^{\ell + 1}).\] In particular, we show that $k = 81632$ is the largest weight for which all the coefficients of $F_{0,k}(z)$ are non-negative. This result has applications to the theory of extremal lattices.