= Solution
For a cusp form $f$, the <invariant norm of a modular form>
$$
y^{k/2}|f(x+iy)|
$$
is modular invariant, bounded on $\mathcal F$, and tends to zero at its cusp. Part a therefore makes it bounded throughout $\mathfrak h$: $|f(x+iy)|\leq Cy^{-k/2}$.
The correct PDF expansion is $f(\tau)=\sum_{n\geq1}a_n(f)q^n$. Fourier inversion gives
$$
a_n(f)e^{-2\pi ny}=\int_0^1f(x+iy)e^{-2\pi inx}\,dx,
$$
and hence
$$
|a_n(f)|\leq Ce^{2\pi ny}y^{-k/2}.
$$
Choosing $y=1/n$ proves the <Fourier coefficient bound for a cusp form> $|a_n(f)|\leq C'n^{k/2}$.
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