Sturm bound for $\Gamma_1(N)$ (reviewed)

The Sturm bound is an upper bound on the least index where the coefficients of the Fourier expansions of distinct modular forms in the same space must differ.

More precisely, for any space $M_k(\Gamma_1(N))$ of modular forms of weight $k$ and level $N$, the Sturm bound is the integer \[ B(M_k(\Gamma_1(N))) := \left\lfloor \frac{km}{12}\right\rfloor,\] where \[ m:=[\SL_2(\Z):\Gamma_1(N)]=N^2\prod_{p|N}\left(1-\frac{1}{p^2}\right). \] If $f=\sum_{n\ge 0}a_n q^n$ and $g=\sum_{n\ge 0}b_n q^n$ are elements of $M_k(\Gamma_1(N))$ with $a_n=b_n$ for all $n\le B(M_k(\Gamma_1(N)))$ then $f=g$; see Corollary 9.19 in [MR:2289048, stein-modforms.pdf] for $k>1$.

The Sturm bound applies, in particular, to newforms of the same level and weight. Better bounds for newforms are known in certain cases (see Corollary 9.19 and Theorem 9.21 in [MR:2289048, stein-modforms.pdf], for example), but for consistency we always take the Sturm bound to be the integer $B(M_k(\Gamma_1(N)))$ defined above.