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

Put $\tau =x+iy$. The modular transformation law andshow that the invariant norm of a modular formis invariant under $\Gamma (1)$. On the region from part (a), it is bounded: it is continuous on every truncated region, while the cusp form condition makes it tend to zero as $y\to \infty$. Thusfor all $x\in \mathbb{R}$ and $y>0$.

The Fourier coefficient formula on one period givesand henceChoosing $y=1/n$ yieldsThis proves the Fourier coefficient bound for a cusp form.