Suppose that $\tau \in \mathbf{H}$ belongs to the complex upper half plane. The quotient $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$ gives an elliptic curve over $\mathbf{C}$. Write this elliptic curve as $E_{\tau}$. We can write $E_{\tau}$ as follows:
$$E_{\tau}: y^2 = 4x^3 - \left( \dfrac{27 j(\tau)}{j(\tau)-1728} \right)x - \left( \dfrac{27 j(\tau)}{j(\tau)-1728} \right),$$
where $j(\tau)$ is the $j$-invariant function. This is a model for the universal elliptic curve $E \to Y_1(N)$ over a modular curve $Y_1(N)$.
In very vague terms, my question is: suppose we have an eigen-cuspform $f \in S_2(\Gamma_1(N))$. Given the differential $\omega = dx/y$ on $E$, how do we compute the "$f$-part" of $\omega$? That is, how do we compute the direct summand of $\omega$ where the Hecke operators act via the Hecke eigenvalues of $f$?
More rigorously: Let $H^0(E, \Omega_{E/Y_1(N)})$ be the space of holomorphic one-forms on $E$ over $Y_1(N)$.
Does the space $H^0(E, \Omega_{E/Y_1(N)})$ of holomorphic one-forms of $E$ have an action of Hecke operators?
If $f$ is a weight $2$ eigenform for $Y_1(N)$, then is there an eigenspace for $H^0(E, \Omega_{E/Y_1(N)})$ where the Hecke operators act via multiplication by the Fourier coefficients of $f$?
If the answer to 2 is yes, then given the differential $w = dx/y \in H^0(E, \Omega_{E/Y_1(N)})$, how do we explicitly compute the component of $\omega$ by which Hecke operators act via the Hecke eigenvalues of $f$?