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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 $k\geq 2$?

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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 $k$.

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