Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations into $$\mathrm{GL}_2(\mathbb{Z}_{\ell} \otimes \mathcal{O}_F)$$ with exceptional set $\ell \mid N$. The standard way to see this is that it is the tensor product of the first $\ell$-adic cohomology of the modular curve of level $N$ with $F$ *over* the Hecke algebra, where we view $F$ as an algebra over the Hecke algebra via $f$.

I read that there is a construction that gives not only the representation but an abelian variety with $\ell$-adic representation equal to that. Could someone please give a readable reference for this? Thanks.