Isogeny decomposition of the Jacobian of a modular curve (awaiting review)

As with any abelian variety, the Jacobian $J_H$ of a modular curve $X_H$ can be decomposed into simple isogeny factors. For modular curves, these simple isogeny factors are modular abelian varieties corresponding to Galois orbits of weight 2 newforms.

We list two types of information about the isogeny decomposition of $J_H$:

• the multiset of dimensions of the simple isogeny factors: $1^3\cdot 2^2$ denotes $3$ factors of dimension $1$ and $2$ factors of dimension $2$;
• the multiset of newforms corresponding to the modular abelian varieties in the isogeny decomposition, listed by label: $\texttt{11.2.a.a}^3\cdot \texttt{13.2.e.a}^2$ denotes five simple isogeny factors, three isogenous to the modular abelian variety corresponding to the newform labelled $\texttt{11.2.a.a}$ and two isogenous to the modular abelian variety corresponding to the newform labelled $\texttt{13.2.3.a}$.

When $X_H$ has genus zero, $J_H$ is the trivial abelian variety of dimension zero, and no isogeny decomposition information is listed.