Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 2 / b / Solution

For a cusp form $f$, the invariant norm of a modular formis modular invariant, bounded on $\mathcal{F}$, and tends to zero at its cusp. Part a therefore makes it bounded throughout $\mathfrak{h}$: $|f(x+iy)|\le C{y}^{-k/2}$.

The correct PDF expansion is $f(\tau )={\sum }_{n\ge 1}{a}_n(f)q^n$. Fourier inversion givesand henceChoosing $y=1/n$ proves the Fourier coefficient bound for a cusp form $|a_n(f)|\le C^{\prime }{n}^{k/2}$.