Lehmer's conjecture for Ramanujan's tau function says that $\tau(n) \neq 0$ for all $n$. In this paper, we generalize D. H. Lehmer's result to give a sufficient condition for level one cusp forms $f$ such that the smallest $n$ for which the Fourier coefficients $a_n(f)=0$ must be a prime. For the unique cusp form $\Delta_{k}$ of level one and weight k with $k=16, 20, 22$, we achieve a large bound $B_k$ of $n$ such that $a_n(\Delta_k)\ne0$ for all $n