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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.

Let $\mathcal{L}$ be the set of complex lattices. A weight-$k$ lattice function is a function satisfyingSet ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ and $f(\tau )=F({\Lambda }_{\tau })$. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, a change of oriented basis gives ${\Lambda }_{\gamma \tau }=(c\tau +d{)}^{-1}{\Lambda }_{\tau }$, henceThus the resulting function has the weight-$k$ transformation law; ordinary invariance under the modular group holds only when $k=0$.

Conversely, a function with this transformation law defines $F({\omega }_2{\Lambda }_{\tau })={\omega }_2^{-k}f(\tau )$ independently of the chosen oriented basis. To obtain actual modular forms, one must additionally require holomorphy on the complex upper half-plane and at the modular cusp; homogeneity alone imposes no analytic regularity. For even $k>2$, the lattice Eisenstein sum is an example, and its scaling law explains its weight.^[ref]

A modular function for a finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a meromorphic function on the complex upper half-plane that is invariant under $\Gamma$ and is meromorphic at every modular cusp:At a modular cusp of width $h$, meromorphicity means that the transformed function has a Laurent series in the local parameter $q_h=e^{2\pi i\tau /h}$ with only finitely many negative powers. Thus a modular function can have poles, in contrast to a modular form, which is required to be holomorphic on the half-plane and at the modular cusps. The Klein j-invariant is a fundamental example for the full modular group.

Some sources use the broader expression modular function of weight $k$ for a meromorphic function satisfyingwith the corresponding meromorphic cusp condition. This is also called a meromorphic modular form; the unqualified term modular function ordinarily means weight zero. Weight-zero modular functions descend to meromorphic functions on the compactified modular curve.^[ref]