Modulo torsion, the Picard group of the quotient of the Siegel upper half plane by a finite-index subgroup of $\text{Sp}_{2g}(\mathbb{Z})$ is just $\mathbb{Z}$. This result should probably be attributed to Borel.

For a calculation of the torsion and explicit line bundles corresponding to the various pieces (plus references), see my paper "The Picard group of the moduli space of curves with level structures", available here. The results you want are discussed starting on page 3.

Modulo torsion, the Picard group of the quotient of the Siegel upper half plane by a finite-index subgroup of $\text{Sp}_{2g}(\mathbb{Z})$ is just $\mathbb{Z}$. This result should probably be attributed to Borel.

For a calculation of the torsion and explicit line bundles corresponding to the various pieces (plus references), see my paper "The Picard group of the moduli space of curves with level structures", available here. The results you want are discussed starting on page 3.