Let $\epsilon\_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G\_\mathbb{Q}$ of finite image and with conductor $N$ prime to $p$, and $F$ be the set of the primes of $F$ above $p$. 


Is the dimension of $H^1\_f(G\_{F,S}, \bar{\mathbb{Q}}\_p(\psi \epsilon^{k}))$ known when $k\geq 2$?



Crystalline extension the $p$-adic cyclotomic character
Yes, this is known.


By Shapiro's lemma we have
$$H^1\_f(G\_{F, S}, \psi \epsilon^k) = H^1\_f(G\_{\mathbf{Q}, S}, \psi \epsilon^k) \oplus H^1\_f(G\_{\mathbf{Q}, S}, \chi\_F \psi \epsilon^k),$$
where $\chi\_F$ is the quadratic character attached to $F$. There are standard formulae for the dimensions of both terms on the right-hand side, in terms of the parity of $k$, which (judging by your previous questions) you already know. So the dimension is $2$ if $k \ge 2$ is even, and $0$ for all other $k$.


