= Ramanujan congruence modulo 691
{c}
{title2=$\tau(n)\equiv\sigma_{11}(n)\pmod{691}$}
The <Fourier expansion of a normalized Eisenstein series> and the one-dimensional weight-twelve <cusp form> space give $691E_{12}=691E_4^3-432000\Delta$. The coefficient of $q^n$, for $n\geq1$, reduces modulo $691$ to $566\sigma_{11}(n)=566\tau(n)$. Since $566$ is invertible modulo $691$, this proves the congruence. Integrality follows from $E_4=1+240\sum\sigma_3(n)q^n$ and the product $\Delta=q\prod_{n\geq1}(1-q^n)^{24}$.
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