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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 $f \in S_1(\Gamma_1(N))$. 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 $Y_1(N)$ and let $f \in S_1(\Gamma_1(N))$ be an eigen-cuspform. Given the differential $w = dx/y \in H^0(E, \Omega_{E/Y_1(N)})$, how do we explicitly compute the $f$-isotypical component of $\omega$ under the action of the Hecke operators?

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 $f \in S_1(\Gamma_1(N))$. 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 $Y_1(N)$ and let $f \in S_1(\Gamma_1(N))$ be an eigen-cuspform. Given the differential $\omega = dx/y \in H^0(E, \Omega_{E/Y_1(N)})$, how do we explicitly compute the $f$-isotypical component of $\omega$ under the action of the Hecke operators?

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 $f \in S_1(\Gamma_1(N))$. 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 $Y_1(N)$ and let $f \in S_1(\Gamma_1(N))$ be an eigen-cuspform. 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 $f$?

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 $f \in S_1(\Gamma_1(N))$. 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 $Y_1(N)$ and let $f \in S_1(\Gamma_1(N))$ be an eigen-cuspform. Given the differential $w = dx/y \in H^0(E, \Omega_{E/Y_1(N)})$, how do we explicitly compute the $f$-isotypical component of $\omega$ under the action of the Hecke operators?