= Maass cusp form
{c}
{title2=$f\in L^2_{\mathrm{cusp}}$}
A <Maass cusp form> is a nonzero <Maass form> with vanishing <constant term of an automorphic form> at every <cusp (automorphic form)>. For weight zero on the <modular group> $\Gamma=\operatorname{SL}_2(\mathbb Z)$, it is a smooth, $\Gamma$-invariant <function> on the <upper half-plane>, square-integrable on a <fundamental domain> with measure $dx\,dy/y^2$, satisfying
$$
\Delta f=\lambda f,\qquad \Delta=-y^2(\partial_x^2+\partial_y^2),\qquad \int_0^1 f(x+iy)\,dx=0\quad(y>0).
$$
The integral condition describes the <cusp (automorphic form)> at infinity; other groups require the corresponding condition in coordinates at each <cusp (automorphic form)>.<Rigorous computation of Maass cusp forms of squarefree level by Andrei Seymour-Howell>[Section 2.1, definition specialized to level one and stated Laplacian sign]{ref}
Translation invariance gives a <Fourier series>. When $\lambda=1/4+r^2$, its decaying modes have the form
$$
f(x+iy)=\sqrt y\sum_{n\ne0}a_n K_{ir}(2\pi |n|y)e^{2\pi inx},
$$
where $K_{ir}$ 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.<Rigorous computation of Maass cusp forms of squarefree level by Andrei Seymour-Howell>[Section 2.4, even and odd Fourier expansions, rewritten in complex exponential notation]{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>.
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