On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties

Let $E$ be an elliptic curve over $\mathbb{Q}$. We prove that the degree of the Shimura parametrization of $E$ arising from the optimal quotient $J^D(M)\to E$ is equal to the $D$-new modular degree of $E$, under mild assumptions on the local Galois representations of $E$. As an application, we prove