Weight 11 compactly supported cohomology of moduli spaces of curves

We study the weight 11 part of the compactly supported cohomology of the moduli space of curves $M_{g,n}$, using graph complex techniques, with particular attention to the case $n=0$. As applications, we prove new nonvanishing results for the cohomology of $M_g$, and exponential growth with $g$, in a wide range of degrees.