How to show $\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$?

Question

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$.

I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$.

I think one can show that for a $k>0$ and $f\in M_k(\Gamma)$ not null in $\mathbb{H}$ f is a cusp form ($\frac{k}{12}$ formula?) and then conclude by induction.

Can somebody give me a hint?