Let $N \in \mathbb{N}$, let $f \in S_2(\Gamma_0(N))$ be a normalised newform, and let $K_f \subset \mathbb{C}$ be the number field generated over $\mathbb{Q}$ by the Fourier coefficients $a_n(f)$ of $f$. Let $\ell$ be a rational prime, and let $\lambda$ be a finite prime of $K_f$ lying above $\ell$, with completion $K_{f,\lambda}$.
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Then there exists an abelian variety $A_f$ over $\mathbb{Q}$, obtained as a quotient of the modular Jacobian $J_0(N)$, together with an embedding
is a two-dimensional [vector space](/page/Vector%20Space) over $K_{f,\lambda}$.
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The natural action of the absolute [Galois group](/page/Galois%20Group) $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $T_\ell A_f$ induces a continuous representation
For every rational prime $p$ with $p \nmid N\ell$, the representation $\rho_{f,\lambda}$ is unramified at $p$, and arithmetic Frobenius $\operatorname{Frob}_p$ satisfies