be a normalized cuspidal Hecke eigenform, so $a_1(f)=1$ and $f$ is an eigenvector for all Hecke operators. Let $\varepsilon:(\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times$ denote the nebentypus character of $f$, equivalently the system of diamond-operator eigenvalues of $f$.
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Then for every prime number $\ell$ and every field embedding $\iota_\ell:\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_\ell$, there exists a continuous semisimple representation
such that $\rho_{f,\ell}$ is unramified at every prime $p \nmid N\ell$, and for every such prime $p$ the characteristic polynomial of an arithmetic Frobenius element $\operatorname{Frob}_p \in G_{\mathbb{Q}}$ is