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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-03-13 18:45:02

**History:**(expand/hide all)

- 2019-03-13 18:45:02 by Andrew Sutherland (Reviewed)
- 2019-03-13 18:42:56 by Andrew Sutherland
- 2019-01-07 16:31:24 by David Roe

**Differences**(show/hide)