# Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through the conjugation action of $D^{x}$ on totally imaginary quaternions, this group is isomorphic to $O(3)$ split away from $p$ and $\infty$. Secondly, through the theory of the class group, a level $\mathfrak{R}$ modular form is associated to a function on the genus of an Eichler order of level $\mathfrak{R}$. Finally, Jacquet-Langlands tells us it is an ordinary modular form, via the trace formula.

Now using the theta series for the Eichler order, we can compute what turns out to be a classical modular form. Does this modular form agree with the one Jacquet-Langlands would give us? It seems to me the answer is yes by strong multiplicity one. Furthermore, if we take the form on $O(3)$, we can take its theta series to get a weight $3/2$ modular form, and apply the Shimura correspondence to get a weight $2$ modular form. (Although Jacquet-Langlands is used in the proof by Waldspurger of this fact, and we do have a nontrivial kernel for some of these maps). Is this the same as the one given by theta series on the Eichler order? I think the answer is no, because many sources say so, but I've not found or constructed a good example yet.