Fourier expansion of Maass form (reviewed)

The Fourier expansion (at infinity) of a Maass form of a non-cocompact subgroup of PSL$(2,\R)$ reads $$f(z)=\sum_{n\in\Z}a(n)\sqrt{y}K_{ir}(2\pi|n|y)e^{2\pi inx},$$ where $K_{ir}(x)$ is a $K$-Bessel function and the expansion coefficients $a(n)$ are related to Hecke eigenvalues.

If the Maass form is even, then the expansion simplifies to $$f(z)=\sum_{n=1}^{\infty}c(n)\sqrt{y}K_{ir}(2\pi ny)\cos(2\pi nx),$$ and if the form is odd $$f(z)=\sum_{n=1}^{\infty}c(n)\sqrt{y}K_{ir}(2\pi ny)\sin(2\pi nx).$$ For a prime $p$ the coefficient $c(p)$ equals the corresponding Hecke eigenvalue.