Non-vanishing Fourier coefficients of modular forms

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 $a_n(f)\mathrm{\ne }0$ for all $n<B$. As examples, we achieve the explicit bounds $B_k$ for the unique cusp form ${\mathrm{\Delta }}_k$ of level one and weight k with $k=16,18,20,22,26$ such that $a_n({\mathrm{\Delta }}_k)\mathrm{\ne }0$ for all $n<B_k$.