Compatibility between base change and Hecke orbits of Hilbert newforms

Let $F/E$ be a Galois extension of totally real number fields, with Galois group $\mathrm{Gal}(F/E)$. Let $\mathfrak{N}$ be an integral ideal which is $\mathrm{Gal}(F/E)$-invariant, and $k \ge 2$ an integer. In this note, we study the action of $\mathrm{Gal}(F/E)$ on the Hecke orbits of Hilbert newforms of level $\mathfrak{N}$ and weight $k$. We also discuss the geometric counterpart to this action, which is closely related to the notion of abelian varieties potentially of $\mathrm{GL}_2$-type. The two actions have some consequences in relation with Langlands Functoriality. We conclude with an example over the maximal totally real subfield $F = \mathbf{Q}(\zeta_{32})^+$ of the cyclotomic field of 32nd root of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and $7$ real places, and $X_0^D(1)$ the Shimura curve attached to $D$. Among other things, our example shows that the field of $2$-torsion of the Jacobian of the curve $X_0^D(1)$ (and its Atkin-Lehner quotient) is the unique Galois extension $N/\mathbf{Q}$ unramified outside $2$, with Galois group the Frobenius group $F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. This completes Noam Elkies' answer~\cite{elk15} to a question posed by Jeremy Rouse on \verb|mathoverflow.net|.