Let $f\in S_2(\Gamma_0(N))$ be a normalized newform with Fourier expansion $f(q)=\sum_{n=1}^{\infty} a_n(f)q^n$ and Hecke field $K_f$. There is an abelian variety $A_f/\mathbb{Q}$, unique up to isogeny, and a Hecke-stable quotient $J_0(N)\to A_f$ such that