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There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3]. What is ...

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 ...