 Solution

The analytic input for these Mellin transforms is elementary cusp decay. For a level-one weight-$k$ cusp form, the invariant norm of a modular form $y^{k/2}|f(x+iy)|$ is invariant under the modular group. It is bounded on the standard fundamental region, because the cusp expansion decays exponentially, and hence bounded on all of $\mathbb{H}$. Fourier inversion givesTaking $y=1/n$ proves the Fourier coefficient bound for a cusp form $c_n=O(n^{k/2})$. In particular rational translates have polynomially bounded coefficients and exponential decay as $y\to \infty$, uniformly in their real parts. These facts justify the initial Mellin integrations and the later entire continuation.