On the rational cohomology of moduli spaces of Prym curves

We investigate the low degree rational cohomology groups of the moduli space of twisted Prym curves ${\overline{Pr}}_{g,n}^{(m_1,\dots ,m_n)}$, where the integer twists $0\le m_i\le 1$ have even sum over $i$. We prove that these groups vanish in odd degree $\le 3$ and that the group in degree $2$ is algebraic. In particular, the results cover the classical moduli spaces of Prym curves and Prym curves with simple ramifications.