= Maass form
{c}
{title2=$\Delta f=\lambda f$}
In the weight-zero convention, a <Maass form> for a discrete subgroup $\Gamma\subseteq\operatorname{SL}_2(\mathbb R)$ is a smooth <function> on the <upper half-plane>, invariant under $z\mapsto(az+b)/(cz+d)$, satisfying a <hyperbolic Laplacian> eigenvalue equation and specified growth conditions at the cusps:
$$
\Delta f=\lambda f,\qquad\Delta=-y^2(\partial_x^2+\partial_y^2).
$$
Additional conditions specify a cuspidal or square-integrable class. The sign of $\Delta$ is a convention and must be stated.<Shifted polyharmonic Maass forms for PSL(2,Z) by Nickolas Andersen and others>[Introduction, depth-one eigenfunction conditions; the paper uses its own Laplacian sign convention]{ref}
The constant <function> $f=1$ is an invariant eigenfunction with $\lambda=0$ and moderate growth, but is not cuspidal: its average along a horizontal period is nonzero. A <Maass cusp form> has zero such constant term at each cusp. The eigenfunction condition allows nonholomorphic <automorphic forms>, connecting arithmetic symmetries with <spectral geometry>.
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