For any p-adic local field K, all 1-dim **semi-stable** Galois repn: $G_K \to Q_p^{*}$ are just $Q_p(n)\otimes \mu$, where $Q_p(n)$ is the Tate twist of cyclotomic character, and $\mu$ an unramified charater.

**My question** is what if we replace the coefficient field to $E \neq Q_p$?

In fact, at the end of the paper by Gerasimos Dousmanis "Rank two filtered $(φ, N)$-modules with Galois descent data and coefficients", the filtered $(\varphi, N)$ modules of all such 1-dim repns are all classified. My question really is, how do we write out the representations explicitly?