Hecke operators on universal elliptic curves 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?
 A: The universal elliptic curve can be written $y'^2 = 4x'^3- g_2 x - g_3$ where $g_2$ and $g_3$ are Eisenstein series. Given any differential on  $E$, simply change coordinates to this family, then divide by $dx'/y'$ to obtain a modular form of weight $1$.
The point is that $g_2$ is modular of weight $4$ and $g_3$ is modular of weight $6$, so if $y$ and $x$ are coordinates in a $\Gamma_1(N)$-invariant presentation of the universal family, then $x/x'$ is a modular function of weight $2$ and $y/y'$ is a modular function of weight $3$. Thus $dx/y $ is equal to $ dx'/y'$ times a modular function of weight $2-3=-1$. Dividing the $\Gamma_1(N)$-invariant equation for your differential form by a modular function of weight $-1$ produces a modular form of weight $1$.
I'm not sure the presentation you gave necessarily works for this, because I think it is only the correct one up to possible quadratic twist. The correct model is the one that actually has a section of order $N$.
Modular forms of weight $1$ certainly don't correspond to differentials on the modular curve. Rather these are modular forms of weight $2$. The relation between differentials on $E$ and differentials on modular curves is simply that they satisfy similar transformation laws under $SL_2(\mathbb Z)$, or, algebro-geometrically, that the line bundle of relative differentials squares to the line bundle of differentials of the base.
