Let $j$ be some integer which is neither 0 nor 1. The Galois cohomology space $H^1({\mathbb Q}_p,{\mathbb Q}_p(j))$ is then 1-dimensional, and thus there is a unique non-split 2-dimensional local Galois representations $\rho_j$ with ${\mathbb Q}_p(j)$ as a sub and ${\mathbb Q}_p$ as a quotient.

I think that $\rho_j$ is crystalline if $j>1$, and not crystalline if $j<1$. (One can just write down all of the 2-dimensional reducible admissible modules, and this should follow...I hope.)

Now take $\rho_j$ and restrict it to the absolute Galois group of ${\mathbb Q}_p(\mu_p^\infty)$. This restricted representation then looks like $\begin{pmatrix} 1 & \psi \newline 0 & 1 \end{pmatrix}$ when $\psi$ is some additive character of $G_{{\mathbb Q}_p(\mu_p^\infty)}$.

My question: Should we be able to explicitly describe this character $\psi$? For instance, is $\psi$ an unramified character?