Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures

We compute the twisted cohomology of the mapping class group with level structures with coefficients the $r$-tensor power of the Prym representations for any positive integer $r$. When $r\ge 2$, the cohomology turns out to be not stable when the genus is large, but it is stable when r is $0$ or $1$. As a corollary to our computations, we prove that the symplectic Prym representation of any finite abelian regular cover of a non-closed finite-type surface is infinitesimally rigid.