Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \mathrm{GL}_2(\mathbb{C})$ be the galois representation attached to $f$.

I have the following question : Let $q$ be a prime ideal of the ring of integers of $F$ such that $q$ is prime to $p$ and $ q \mid N$:

• If $U_q (f)=a_q.f$ with $a_q \ne 0$. Is it equivalent to the fact that there exits a line of $\mathbb{C}^2$ fixed by the inertia group $I_q$ and on which $\mathrm{Frob}_q$ acts via $a_q$ ?

• If $U_q(f)=0$. Is it equivalent to the fact that $\rho^{I_q}=0$ ?