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 $k\geq 2$?