 Fourier coefficient growth criterion for a level-one cusp form

Let even $k\ge 4$ and let $g(\tau )={\sum }_{n\ge 0}{b}_n{q}^n$ 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.