Let $p$ be a prime number, let $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute [Galois group](/page/Galois%20Group) of $\mathbb{Q}$, and let
for every complex conjugation element $c \in G_{\mathbb{Q}}$, and for $p = 2$ the oddness condition is interpreted in the standard mod $2$ sense.
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Then $\bar{\rho}$ is modular: there exist a normalized cuspidal Hecke eigenform $f$, a number field $K_f$ generated by the Hecke eigenvalues of $f$, and a finite place $\lambda$ of $K_f$ with residue characteristic $p$ such that