# Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$.

I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$.

I have already shown, that $h$ is invariant under $\Gamma$. I think that it's sufficient, that $h$ is bounded on the standard fundamental domain $\mathscr{F}$, although I have not proven that.

Hint: Since $h$ is continuous, you simply have to show that it stays bounded as $y$ tends to infinity (uniformly in $x$ -- here $z=x+iy$). Consider the expansion $f(z) = \sum_{n\geq 1} a_n {\rm e}^{2\pi i n z}$, which starts at $n=1$ since $f$ is cuspidal. As $y\to\infty$, what happens for each terms in the series ? How fast do they decay ?