**Short version**: is there a canonical way to adelize a classical Hecke eigenform automorphic form when the adelic quotient has many components? If not, what are the different "choices", how many, etc.?

**Some sketched details**: Let $A$ be a central simple algebra over a number field $F$, e.g. $A$ is the matrix algebra over $F$. Let $\mathcal{O} \subset A$ be an order, e.g. the matrix ring over the ring of integers $\mathfrak{o}$ of $F$. The adelic quotient $A^\times \backslash \hat{A}^\times / \hat{\mathcal{O}}^\times$ is (assuming some Eichler condition) a disjoint union parametrised by the class group $F^\times_{>0} \backslash \hat{F}^\times / \operatorname{nr}(\hat{\mathcal{O}}^\times)$ (see Thm. 28.5.5 in Voight's Quaternion Algebras). This class group can be non-trivial if $F$ has non-trivial narrow class group or if the order $\mathcal{O}$ is small, more precisely if the image of the local norm $\operatorname{nr}(\mathcal{O}^\times_\mathfrak{p})$ fails to be the whole unit group $\mathfrak{o}_\mathfrak{p}^\times$, where $\mathfrak{p}$ is a prime of $F$.

Each such component corresponds to a locally symmetric space (e.g. for quaternion algebras this would be an arithmetic quotient of the upper half plane). In this way, each adelic automorphic form is given by a tuple of classical automorphic forms on these components (see e.g. Shimura's 1978 Hilbert modular forms paper).

If one has a classical automorphic form on one of these components and we assume that it is also a Hecke eigenform, then is there a canonical way to choose forms on the other components so that the corresponding adelic form is a Hecke eigenform. Of course, Hecke eigenform here has two meanings, referring to the classical and to the global Hecke algebra accordingly. Are there any references for this?