Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic representations of $PGL(2)$ are representations of $GL(2)$ with trivial central character, and Jacquet-Langlands describes the ones that come from representations of $D^{\times}$ where $D$ is a definite quaternion algebra. (Supposedly: I've not found where in the book that result is) Then using the fact that $D^{\times}$ is a double cover of the nonsplit $SO(3)$, we should be able to prove Birch's conjecture. However, this requires a description of the representations of $D^{\times}$ that includes the central characters as a representation of $D^{\times}$. Does this appear somewhere in Jacquet-Langlands, or do people have some other source? Or is this line of reasoning insufficient to get Birch's conjecture?

In the case of squarefree levels we can avoid supercuspidals, so a description that only works for the principal series and the Steinberg (and twists of Steinberg) suffices.