I've been reading A first course in modular forms by F.Diamond and J.Shurman. In chapter I, the problem 1.2.6. says:
Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$, thus containing $\Gamma(N)$ for some $N$, and suppose that the function $f:\mathcal{H}\to\mathbb{C}$ is holomorohic and weight-k invariant under $\Gamma$. Suppose also that in the Fourier expansion $f(\tau)=\sum_{n=0}^{\infty}a_nq_N^n$, the coefficients for $n>0$ satisfy $|a_n|\leq Cn^r$ for some positive constants $C$ and $r$.
I want to know if that estimation is always feasible.
See p.213-216. The idea is that if $f(z)$ is a weight-$k$ cusp form then $|f(z)| (\Im(z))^{k/2}$ is $\Gamma$ invariant and continuous on $\Gamma \setminus \mathcal{H}^*$, hence it is bounded on $\Im(z) > 0$, say by $C$.
Also for every $y >0$ and hence for $y=1/n $ : $$a_n = \int_{iy}^{iy+1} f(z) e^{-2i \pi nz}dz = n^{k/2}\int_0^1 f(x+i/n) (\Im(x+i/n))^{k/2} e^{-2i \pi n(x+i/n)}dx \le e^{2\pi} C n^{k/2}$$
If $f(z)$ is not a cusp form, then substract the weight-$k$ Eisenstein series whose coefficients are $\mathcal{O}(n^r)$ (with $r = k-1$)