The symmetry of a Maass form (reviewed)

The modular group and its congruence subgroups has an outer automorphism of order two, which is given by the reflection in the imaginary axis: \( z=x+iy \mapsto -\bar{z}=-x+iy \). This map belongs to $\mathrm{PGL}(2,\mathbb{Z})$ and can be represented by the matrix $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

A Maass form $f$ is said to be even if $f(-\bar{z})=f(z)$ and odd if $f(-\bar{z})=-f(z)$.