The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$

We prove that the rational cohomology group $H^{11}({\bar{\mathcal{M}}}_{g,n})$ vanishes unless $g=1$ and $n\ge 11$. We show furthermore that $H^k({\bar{\mathcal{M}}}_{g,n})$ is pure Hodge-Tate for all even $k\le 12$ and deduce that $\mathrm{\#}{\bar{\mathcal{M}}}_{g,n}({\mathbb{F}}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}({\bar{\mathcal{M}}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.