For a continuous irreducible representation


$\rho: G\_{\mathbb{Q}\_p}\rightarrow GL\_n(\overline{\mathbb{Q}\_p})$,


is it possible for both $D\_{cris}(\rho)$ and $D\_{cris}(\chi\otimes\rho)$ to be nonzero, where $\chi$ is some non-crystalline character? 



Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$?
Let me use Colmez' article "Representations triangulines" as a reference. Let $V$ be a repn which satisfies your condition.


By proposition 4.3, $V$ is trianguline. By proposition 4.10, the HT weight of $\chi$ has to be an integer. You can then assume that $\chi$ has finite order, and this implies that $V$ is potentially crystalline on an abelian extension of $Q\_p$ (aka crystabelline).


Conversely, it seems likely that one can give examples of irreducible crystabelline representations $V$ such that $D\_{cris}(V)$ and $D\_{cris}(V \otimes \chi)$ are both nonzero, with $\chi$ of finite order (see the examples in 2.4 of Berger-Breuil's "Sur quelques representations potentiellement cristallines de $GL\_2(Q\_p)$").


