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The constant term of an automorphic form is its average along the periodic or unipotent direction associated with a cusp. In the basic period-one upper-half-plane setting,It is the zeroth coefficient of the Fourier series in $x$. Despite the name, it may depend on $y$. For a cusp width $h$, use the normalized average $h^{-1}{\int }_0^h$ after moving the chosen cusp to infinity.^[ref]

For a weight-zero Maass form with $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$ and $\Delta f=\lambda f$, periodicity makes the averaged $x$-derivative term vanish. ThusIf $\lambda =s(1-s)$ and $s\ne 1/2$, the solutions are $A{y}^s+B{y}^{1-s}$. At $s=1/2$, the repeated root gives $y^{1/2}(A+B\log y)$. A constant Fourier mode need not therefore be a constant function on the upper half-plane.

For the constant automorphic function $f=1$, the constant term is one and $\lambda =0$. A Maass cusp form instead has identically zero constant term at every cusp. The condition concerns the whole coefficient function, rather than merely its value at one height.

For a finite-index subgroup $\Gamma$ of ${\mathrm{SL}}_2(\mathbb{Z})$, a cusp is a $\Gamma$-equivalence class of rational boundary points, including infinity:Geometrically it describes a noncompact end of the quotient of the upper half-plane. Adding these classes compactifies the corresponding modular curve. The added points describe boundary data of the quotient.^[ref]

The full modular group has one cusp: a reduced rational number $a/c$ is the image of infinity under a matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with determinant one. In contrast, ${\Gamma }_0(2)$ has inequivalent cusps represented by infinity and zero. A matrix sending infinity to zero must have $a=0$, hence $-bc=1$ and odd $c$, so it cannot belong to ${\Gamma }_0(2)$.

At infinity the cusp width is the least positive $h$ for which $z\mapsto z+h$ belongs to the effective stabilizer. The local coordinate $q=e^{2\pi iz/h}$ tends to zero as $y\to \infty$. Filling in $q=0$ adds one point. At other cusps one first changes coordinates by a matrix carrying infinity to the chosen rational boundary point. Growth and constant term of an automorphic form conditions must be checked at each inequivalent cusp.

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.

A shifted polyharmonic Maass form is a smooth automorphic function that satisfies a generalized eigenfunction equation for a hyperbolic Laplacian. In weight zero, choose the convention $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$ and requirefor an integer $m\ge 1$, together with automorphy and the specified growth conditions at each cusp. Moderate growth means at most polynomial growth in cusp coordinates. The weight-$k$ version replaces $\Delta$ by the corresponding Weight-k hyperbolic Laplacian and uses the weight-$k$ transformation law.^[ref]

The least possible $m$ is the exact depth. For $m=1$ this is an ordinary eigenfunction condition; higher depth allows a chain of generalized eigenfunctions. It is not merely an equation with a different eigenvalue. The cited paper uses the opposite Laplacian sign, so its eigenvalue parameter is the negative of the one used here.

For an illuminating construction, let a nonholomorphic Eisenstein series $E(z,s)$ satisfy $\Delta E=s(1-s)E$. At a regular value $s=2$, differentiate in $s$. The function $F={\partial }_{s}E{|}_{s=2}$ obeysThe first right-hand side is nonzero, so this gives depth two. Parameter differentiation introduces a generalized eigenfunction while preserving automorphy.^[ref]