Theorem 63.31.2. Given an eigenform $f$ with values in $\overline{\mathbf{Q}}_ l$ and eigenvalues $u_ v\in \overline{\mathbf{Z}}_ l^*$ then there exists

continuous, absolutely irreducible where $E$ is a finite extension of $\mathbf{Q}_\ell $ contained in $\overline{\mathbf{Q}}_ l$ such that $t_ v = \text{Tr}(\rho (F_ v))$, and $u_ v = q_ v^{-1}\det \left(\rho (F_ v)\right)$ for all places $v$.

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