Non-vanishing Fourier coefficients of modular forms

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 $n<B_k$.