The Picard group of the moduli space of curves with level structures - [pdf] [ps]
Duke Math. J. 161 (2012), no. 4, 623–674.
Computer code used in paper here.

Abstract :
For \$4 \nmid L\$ and \$g\$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level \$L\$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level \$L\$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level \$L\$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod \$L\$ symplectic group with coefficients in the adjoint representation.