Let $H$ be a totally complex Galois extension of $\mathbb{Q}$ and $g:G_H \rightarrow \bar{\mathbb{Q}}_p$ be a continuous morphism. By class field thoery we have $\mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\simeq \mathrm{Hom}((O_H \otimes \mathbb{Z}_p)^{\times}/\bar{O_H^{\times}},\bar{\mathbb{Q}}_p)$. It is known that the $p$-adic logarithms $\log_p(h(.))$ where $h \in \mathrm{Gal}(H/\mathbb{Q})$ rise to a basis of $\mathrm{Hom}((O_H \otimes \mathbb{Z}_p)^{\times},\bar{\mathbb{Q}}_p) \simeq \mathrm{Hom}(\prod_{w\mid p} I_w,\bar{\mathbb{Q}}_p) $ where $I_w$ is the inertia at $w$ which is a prime of $H$ above $p$.
Assume that $g(.)=\sum a_h \log_p(h(.))$ where $h \in \mathrm{Gal}(H/\mathbb{Q})$ and $G_{\mathbb{Q}_p} \subset G_H$. Is there some method to compute $g(\sigma_p)$ where $\sigma_p$ is a lift of the Frobenius at $p$?
