Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations: $$ t\mapsto \begin{pmatrix} \xi(t) & \\ & \xi'(t)\end{pmatrix} $$ for quasi-characters $\xi, \xi':F^{\times} \to \mathbb{C}^{\times}$, or $$ t\mapsto \xi(t) \begin{pmatrix} 1& v(t)\\&1\end{pmatrix} $$ for some quasi-character $\xi$ and a valuation map $v:F^{\times} \to \mathbb{Z}$.
I want to know the complete classification of every (finite dimensional) admissible representation of $F^{\times}$ (including reducible ones). It seems that the above representations like Jordan blocks, so maybe all the finite dimensional representations look like the above ones. Is this true?