A Maass form for a congruence subgroup $\Gamma$ of $SL(2, \mathbb Z)$ is a complex valued smooth function $f$ on the upper half plane $\mathbb{H}$ such that
- $f$ is $\Gamma$-invariant
- $f$ is eigenvalue of the Laplacian
- $f$ is of at most polynomial vertical growth
These correspond to the even principal series $P_0$. I would like to know if we have a classical/explicit definition for the odd Maass forms (i.e. the functions corresponding to off principal series). I don't find much references on the topic.