The next theorem is known as Sturm's bound.
Theorem:Let $\mathfrak{m}$ be a prime ideal in the ring of integers $\mathcal{O}$ of a number field $K$, and let $\Gamma$ be a congruence subgroup of of index $m$ and level $N$. Suppose $f\in M_k(\Gamma,\mathcal{O})$ is a modular form and \begin{align} \mathrm{ord}_{\mathfrak{m}}(f)>\dfrac{km}{2}. \end{align} Then $f\equiv 0\;(\mathrm{mod}\,\mathfrak{m})$.
I want to know if all the sophisticated techniques that Sturm uses in his proof are necessary? In other words, is there a elementary proof? I found, in the case of $\mathrm{SL}_2(\mathbb{Z})$, a sketch of a proof in a book (Problems In The Theory of modular forms by M. Ram Murty ) but i can't realize a complete proof. Especifically, I'm struggling with the width of cusps, why the existence of a represent of a coset associated to a cusp with $h$ as the value of the width implies the existence of other $h-1$ representatives with the same value of width.