I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta characteristic) $\kappa$ on $C$ via the Abel-Jacobi map $\alpha_c$ based at $c \in C$.

I know (from looking at Birkenhake and Lange 11.3) that by Riemann's Theorem and the Seesaw principle

$\alpha_c^*\mathcal{O}_J(\Theta _{\kappa}) = \kappa \otimes \mathcal{O}_C(c)$,

where $\Theta _{\kappa}$ is the (symmetric) theta divisor that $\kappa$ determines by $\alpha _{\kappa}^* \Theta _{\kappa} = W _{g-1} \subset Pic^{g-1}$. So what I guess I'm looking for is a way to move the $c$ dependence over to the left side of this equation. For what it's worth, my hope is to be able to lift the vector bundle $\kappa \oplus \kappa^{-1}$ to $J$, which I'm assuming is just a matter of lifting $\kappa$.

This doesn't seem difficult but I don't have a lot of experience with Abelian varieties and I'm not sure how to get started.