# Explicit Jacquet-Langlands correspondence

Jacquet-Langlands correspondence gives a 1-to-1 correspondence between automorphic forms on $$\mathrm{GL}_{2}(\mathbb{Q})$$ and automorphic forms on $$\mathrm{GL}_{1}(D)$$, where $$D$$ is a division algebra over $$\mathbb{Q}$$. As one can consider (holomorphic) modular forms as $$\mathrm{GL}_{2}(\mathbb{Q})$$ automorphic forms, I think there should exists some kind of automorphic functions on $$\mathrm{GL}_{1}(D)$$ corresponds to the modular form. However, I can't imagine how the automorphic function on division algebra should look like. According to this note, the corresponding function might be a function on $$\mathfrak{h}^{n}$$, where $$\mathfrak{h}$$ is a complex upper half plane and $$n$$ is the number of infinite places which $$D$$ split, with suitable transformation laws. Could anyone give more detailed explanations or references in this direction? For example, what kind of function may corresponds to the discriminant form $$\Delta(z) \in S_{12}(1)$$?

Adelically, ($$L^2$$-)automorphic forms on (the multiplicative group of) a division algebra $$D/F$$ are just elements of $$L^2(G(F) \backslash G(\mathbb A_F))$$, for a number field $$F$$. Since you're working over $$F = \mathbb Q$$, the number of infinite places $$D$$ is split is 0 or 1, according to whether $$D \otimes \mathbb R$$ is Hamilton's quaternions or $$M_2(\mathbb R)$$.
Note that if you want a newform $$f \in S_k(N)$$ to correspond an automorphic form $$\phi$$ on a quaternion algebra $$D$$, it is necessary $$D$$ ramifies at each $$p | N$$, so level 1 forms do not transfer to (non-split) quaternion algebras.