Non-vanishing Fourier coefficients of modular forms

TL;DR

This paper extends Lehmer's result to level one cusp forms with integral Fourier coefficients, providing conditions under which the first zero coefficient occurs at a prime and presenting a method to compute bounds for non-vanishing coefficients.

Contribution

It generalizes Lehmer's theorem to a broader class of cusp forms and introduces a computational method for explicit bounds on non-vanishing Fourier coefficients.

Findings

Established a sufficient condition for the first zero Fourier coefficient to be at a prime

Developed a method to compute bounds for non-vanishing Fourier coefficients

Explicit bounds computed for specific cusp forms of levels and weights

Abstract

In this paper, we generalize D. H. Lehmer's result to give a sufficient condition for level one cusp forms $f$ with integral Fourier coefficients such that the smallest $n$ for which the coefficients $a_n(f)=0$ must be a prime. Then we describe a method to compute a bound $B$ of $n$ such that $a_n(f)\ne0$ for all $n<B$. As examples, we achieve the explicit bounds $B_k$ for the unique cusp form $\Delta_{k}$ of level one and weight k with $k=16, 18, 20, 22, 26$ such that $a_n(\Delta_k)\ne0$ for all $n<B_k$.