 Maass cusp form

A Maass cusp form is a nonzero Maass form with vanishing constant term of an automorphic form at every cusp. For weight zero on the modular group $\Gamma ={\mathrm{SL}}_2(\mathbb{Z})$, it is a smooth, $\Gamma$-invariant function on the upper half-plane, square-integrable on a fundamental domain with measure $dxdy/y^2$, satisfyingThe integral condition describes the cusp at infinity; other groups require the corresponding condition in coordinates at each cusp.^[ref]

Translation invariance gives a Fourier series. When $\lambda =1/4+r^2$, its decaying modes have the formwhere $K_{ir}$ is a Modified Bessel function of the second kind. The absent $n=0$ term expresses cuspidality. The Fourier coefficients must also satisfy the full automorphic symmetry.^[ref]

The constant function $f=1$ is a useful test: it satisfies the symmetry and has eigenvalue zero, but its horizontal average is $1$, so it is not a Maass cusp form. The extra condition isolates oscillating eigenfunctions connecting arithmetic symmetry with spectral geometry.