It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic of order 12 generated by $\nu$ coming from the Hodge-de Rham sequence of the relative cohomology sheaf $H^1_{dR}(\mathcal{E}\to \mathcal{M}_{ell})$. It is also a somewhat folklorish "fact" that $\nu$ is related to the obstruction to the existence of a "true" modular holomorphic Eisenstein series $E_2$. One way I know to make this last vague claim slightly more contentful is that in the Hodge-de Rham spectral sequence, $\nu$ survives and becomes the Euler class in $H^2(\mathcal{M}_{ell},\mathbb{Z}) \cong \mathbb{Z}/6$, and $E_2$ "wants" to be a kind of transgression of the Euler class, c.f. the polylogarithmic construction of Eisenstein series.

On the other hand, one very concrete but seemingly meaningless way to make this precise is to say that we have for $\gamma = \begin{pmatrix}a&b\\c&d\end{pmatrix}\in \text{SL}_2(\mathbb{Z})$ that
$$E_2(\tau)|_\gamma - E_2(\tau)=\frac{12c}{c\tau +d}$$
where the action of $\gamma$ is the usual weight-$2$ action and $E_2$ is the holomorphic quasimodular weight-$2$ Eisenstein series. This looks very suggestive; it appears to say that the coboundary of the function $E_2$, living in the space $\mathcal{O}_2$ of weight-two functions on the upper-half plane, demonstrates that $12$ times the cocycle $c/(c\tau+d)$ is zero in cohomology. Of course, this is gibberish, since $\mathcal{O}_2$ is a complex vector space - and to be honest, I can't imagine there's any way to make a Hopf algebroid presentation for $\mathcal{M}_{ell}$ over the integers using the action of $\text{SL}_2(\mathbb{Z})$ on functions on the upper half-plane. But nevertheless that $12$ is suggestive enough that it's been bothering me for a long time, so I'm finally breaking and asking, **is there some way that this nonsense can be made into a precise statement?**

I ask this because I'm very curious about the existence of similar explicit forms for other coherent extension classes, e.g. the generator of $H^1(\mathcal{M}_{ell},\omega)\cong \mathbb{Z}/2$.