I'm stuck with p.238 of Diamond's book on modular forms. If $\{f_j\}_{j=1}^g$ is the eigenform basis of $S_2(\Gamma_0(N))$, fixing $p_0 \in X_0(N)$ we have an holomorphic map $$\phi : X_0(N) \to J(X_0(N)) = \mathbb{C}^g/\Lambda, \qquad \phi(p) = \left(\int_{p_0}^p f_1(z)dz, \ldots, \int_{p_0}^p f_1(z)dz\right)$$ where $\Lambda \simeq \mathbb{Z}^{2g}$ is the image of the closed loops in $X_0(N)$.
Question : can you explain how the Hecke operators $ \langle d \rangle,T_n: S_k(\Gamma_0(N)) \to S_k(\Gamma_0(N))$ act on the Jacobian variety $J(X_0(N))$ ?
Edit : Milne p.150 explains it much better, in such a way that it is obvious $T_p \Lambda \subseteq \Lambda$, and assuming they are equal, we get an action of $T_p on $J(X_0(N))$