  Let $p\geq 5$ be a prime number, $\mathbb{F}$ a finite field of
characteristic $p$ and let $\bar{\chi}$ be the mod-$p$ cyclotomic character.
Let $\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow
\operatorname{GL}_2(\mathbb{F})$ be a Galois representation such that the local
representation $\bar{\rho}_{\restriction \operatorname{G}_{\mathbb{Q}_p}}$ is
flat and irreducible. Further, assume that
$\operatorname{det}\bar{\rho}=\bar{\chi}$. The celebrated theorem of Khare and
Wintenberger asserts that if $\bar{\rho}$ satisfies some natural conditions,
there exists a normalized Hecke-eigencuspform $f=\sum_{n\geq 1} a_n q^n$ and a
prime $\mathfrak{p}|p$ in its field of Fourier coefficients such that the
associated $\mathfrak{p}$-adic representation ${\rho}_{f,\mathfrak{p}}$ lifts
$\bar{\rho}$. In this manuscript we prove a refined version of this theorem,
namely, that one may control the valuation of the $p$-th Fourier coefficient of
$f$. The main result is of interest from the perspective of the $p$-adic
Langlands program.
