This is a generalisation of my earlier question about generators for the level 1 Hecke algebra.
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. There's an explicit bound $S$ (the "Sturm bound") known such that if $f \in M_k(\Gamma)$ (the space of modular forms of weight $k$ and level $\Gamma$) satisfies $a_n(f) = 0$ for $0 \le n \le S$, then $f = 0$, where $a_n(f)$ denote the coefficients of the $q$-expansion of $f$ (at the cusp $\infty$).
Since $M_k(\Gamma)$ is finite-dimensional and doesn't contain any nonzero constant functions, there must actually exist some $S'$ such that if $a_n(f) = 0$ for $1 \le n \le S'$ (but with no assumption on $a_0(f)$), then in fact $f = 0$. Can one give an effective upper bound on $S'$ (in terms of $k$ and the index $[\operatorname{SL}_2(\mathbb{Z}) : \Gamma]$)?
I'd be interested to know the answer to this even for the special case $\Gamma = \Gamma_0(N)$ or $\Gamma_1(N)$. For these groups it is equivalent to asking that the Hecke algebra acting on $M_k(\Gamma)$ is spanned by the operators $T_1, \dots, T_{S'}$ (hence the title of the question).
