Fourier coefficient growth criterion for a level-one cusp form
= Fourier coefficient growth criterion for a level-one cusp form
Let even $k\geq4$ and let $g(\tau)=\sum_{n\geq0}b_nq^n$ be a level-one modular form of weight $k$. If $b_n=O(n^{k/2})$, then $b_0=0$ and $g$ is a cusp form. Otherwise subtract the matching Eisenstein series; its coefficients grow like $\sigma_{k-1}(n)$, contradicting the bound at prime indices.