  Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\Gamma$ be the
Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a
classification of $p$-adic representations of
$\mathrm{Gal}\left(\overline{K}/K\right)$ in terms of
$(\varphi,\Gamma)$-modules. A useful aspect of this classification is Berger's
dictionary which expresses invariants coming from $p$-adic Hodge theory in
terms of these $\left(\varphi,\Gamma\right)$-modules. In this paper, we use the
theory of locally analytic vectors to generalize this dictionary to the setting
where $\Gamma$ is the Galois group of a Lubin-Tate extension of $K$. As an
application, we show that if $F$ is a totally real number field and $v$ is a
place of $F$ lying above $p$, then the $p$-adic representation of
$\mathrm{Gal}\left(\overline{F}_{v}/F_{v}\right)$ associated to a finite slope
overconvergent Hilbert eigenform which is $F_{v}$-analytic up to a twist is
Lubin-Tate trianguline. Furthermore, we determine a triangulation in terms of a
Hecke eigenvalue at $v$. This generalizes results in the case $F=\mathbb{Q}$
obtained previously by Chenevier, Colmez and Kisin.
