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 $a_n(f)\ne 0$ 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 $a_n({\Delta }_k)\ne 0$ for all $n<B_k$.

We add the case of $k=18$ in this revision. For the weight 16 case, we have one more polynomial which could be used to improve the bound in this case

References in corpus (1)

• On the computation of Galois representations associated to level one modular forms