Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 23 / 5 / a

For a cusp form, the invariant norm of a modular form $|f(\tau )|(\mathrm{Im}\tau {)}^{k/2}$ is bounded on the entire half-plane. It is invariant under the level group; on a truncated fundamental region of a modular subgroup boundedness is compactness, and near each modular cusp exponential decay of the modular cusp expansion dominates the power of height. With this bound, Fourier inversion on a horizontal interval givesChoosing $y=1/n$ proves the Fourier coefficient bound for a cusp form, $a_n=O(n^{k/2})$. Therefore $\sum |a_n|n^{-\sigma }$ converges when $\sigma >k/2+1$, and uniformly on every compact subset of that half-plane. The locally uniform limit of its holomorphic terms is holomorphic, so