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We compute the twisted cohomology of the mapping class group with levelstructures, coefficients being $r$-tensor power of the Prym representations forany positive integer $r$. When $r\ge 2$, the cohomology is not stable when thegenus is large, unlike the rational cohomology of the mapping class group withlevel structures. As a corollary to our computations, we prove that the Prymrepresentation of any finite abelian cover of a non-closed finite-type surfaceis locally rigid.

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