Sturm bound (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 $M_k(N,\chi)$ must differ.

More precisely, for any space $M_k(N,\chi)$ of modular forms of weight $k$, level $N$, and character $\chi$, the Sturm bound is the integer \[ B(M_k(N,\chi)) := \left\lfloor \frac{km}{12}\right\rfloor,\] where \[ m:=[\SL_2(\Z):\Gamma_0(N)]=N\prod_{p|N}\left(1+\frac{1}{p}\right). \] If $f=\sum_{n\ge 0}a_n q^n$ and $g=\sum_{n\ge 0}b_n q^n$ are elements of $M_k(N,\chi)$ with $a_n=b_n$ for all $n\le B(M_k(N,\chi))$ then $f=g$; see Corollary 9.20 in [MR:2289048, stein-modforms.pdf] for $k>1$ and Lemma 5 in [arXiv:1605.05346] for $k=1$.

The Sturm bound applies, in particular, to newforms of the same level, weight, and character. 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 $B(M_k(N,\chi))$ defined above.

Note that the Sturm bound for $S_k^{\mathrm{new}}(N,\chi)$ does not apply (in general) to the space \[ S_k^{\mathrm{new}}(N,[\chi]):= \bigoplus_{\chi'\in [\chi]}S_k^{\rm new}(N,\chi') \] associated to the Galois orbit $[\chi]$; rather, it applies to each direct summand $S_k^{\rm new}(N,\chi')$.