Theorem 63.31.3. Suppose $\mathbf{Q}_ l \subset E$ finite, and

$\rho : \pi _1(X)\to \text{GL}_2(E)$

absolutely irreducible, continuous. Then there exists an eigenform $f$ with values in $\overline{\mathbf{Q}}_ l$ whose eigenvalues $t_ v$, $u_ v$ satisfy the equalities $t_ v = \text{Tr}(\rho (F_ v))$ and $u_ v = q_ v^{-1}\det (\rho (F_ v))$.

Proof. See [D1]. $\square$

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