= 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 gives
$$
c_n=e^{2\pi ny}\int_0^1f(x+iy)e^{-2\pi inx}\,dx.
$$
Taking $y=1/n$ 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.
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