= Constant term of an automorphic form
{title2=$a_0(y)=\int_0^1 f(x+iy)\,dx$}
The <constant term of an automorphic form> is its average along the periodic or unipotent direction associated with a <cusp (automorphic form)>. In the basic period-one upper-half-plane setting,
$$
a_0(y)=\int_0^1 f(x+iy)\,dx.
$$
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.<Modular Functions and Modular Forms by James S. Milne>[sections 2 and 4, cusp coordinates and Fourier expansions]{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. Thus
$$
-y^2a_0''(y)=\lambda a_0(y).
$$
If $\lambda=s(1-s)$ and $s\ne1/2$, the solutions are $Ay^s+By^{1-s}$. 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.
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