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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 $f(-\bar{z})=-f(z)$.

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• Review status: reviewed
• Last edited by David Farmer on 2019-05-01 11:24:23
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