Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 1 / c

Suppose for a contradiction that $b_0\ne 0$. The Eisenstein series in the question has constant term $2\zeta (k)$, sohas zero constant term and is therefore a level-one cusp form of weight $k$. Part (b) gives the coefficient bound $[q^n]h=O(n^{k/2})$. Since $b_n=O(n^{k/2})$ as well, the displayed Fourier expansion of $G_k$ would imply

Take $n=\ell$ through the primes. Thenwhich cannot be $O({\ell }^{k/2})$ because $k-1>k/2$ for $k\ge 4$. Therefore $b_0=0$, so $g$ vanishes at the only cusp of $\Gamma (1)$ and belongs to $S_k(\Gamma (1))$. This is the Fourier coefficient growth criterion for a level-one cusp form.