What is the explicit eigenvalues of Hilbert modular forms?

Question

Let $F$ be a totally real number field and let $I$ denote the set of embeddings $\tau:F\to \mathbb{R}.$ Let $k=(k_\tau)\in\mathbb{Z}^I_{>0}$ and suppose all the $k_\tau$'s have the same parity. Let $f$ be a Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ (an ideal of $\mathcal{O}_F$). It is well known that for a ring $\mathcal{O}$ of integers of a suitable $p$-adic local field, there is a Galois representation $$\rho_f:Gal(\bar{F}/F)\to GL_2(\mathcal{O})$$ such that for $\mathfrak{q}\nmid\mathfrak{n}p$, we have $$trace(Frob_\mathfrak{q})=\theta(T_\mathfrak{q}),det(Frob_\mathfrak{q})=\theta(S_\mathfrak{q})N_{F/\mathbb{Q}}(q)$$ where $T_\mathfrak{q},S_\mathfrak{q}$ are Hecke operators. My question is can we write $\theta(S_\mathfrak{q})$ explicitly? This means if we can use $k$ and $N_{F/\mathbb{Q}}(\mathfrak{q})$ and the character of $f$ to write it out. For example, if $f\in S_k(\Gamma_0(N),\omega)$ is a classical modular form over $\mathbb{Q}$, then $\theta(S_q)=\omega(q)q^{k-2}$.

Answer 1

If $\pi$ is a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_F)$ of level $\mathfrak{N}$ and such that $\pi_{\infty}$ is a holomorphic discrete series of weight $(k,w_0)$, then your $\psi$ is the central character of the automorphic form $\pi$ and it is a Hecke character of weight $-w_0.t$, in the sense that $\psi |.|^{w_0}_{\mathbb{A}_F}$ has a finite order.