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"](http://arxiv.org/abs/0711.2137), 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? 



1-dimensional semi-stable Galois representations with coefficients
If $E \neq Q\_p$ then there may be more $1$-dimensional crystalline representations that the ones you mention. By Lubin-Tate theory, every character of $G\_K$ can be written as an unramified character times a character of $O\_K^\times$ (after making proper choices and identifications). The algebraic characters of $O\_K^\times$ are then crystalline and if $E$ contains $K$, then these provide examples of crystalline characters.


EDIT : oops sorry, I did not read the question carefully enough, I did not see that you were asking for an explicit description of *all* such representations. As David Loeffler pointed out, the answer is in Brian Conrad's paper (now in appendix B, the paper having grown since last time I mentioned it). 


Here is what it basically says, using the same identifications as above: assume that $E$ contains $K^{gal}$ (it's usually harmless to assume that the coefficient field is large enough). If $s$ runs through the set of embeddings $s : K \to E$ and the $a\_s$ are integers, then $x \mapsto \prod\_s s(x)^{a\_s}$ gives rise to a crystalline character of $G\_K$ and they're all of this type times an unramified character.


