  We compute the Picard group of the universal abelian variety over the moduli
stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over
$\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove
that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a
suitable form for $\mathscr A_{g,n}$.
