Maass form coefficients (reviewed)

Let $f$ be a Maass newform of weight 0 on $\Gamma_0(N)$ with spectral parameter $R$. If $f$ has even symmetry then as a function on the complex upper half plane it can be uniquely expressed in the form \[ f(x+iy) = \sum_{n=1}^\infty a_n\sqrt{y}K_{iR}(2\pi ny)\cos(2\pi nx), \] where $K_{iR}(\cdot)$ denotes the $K$-Bessel function with parameter $\nu=iR$ and the $a_n$ are real numbers.

If $f$ has odd symmetry it can instead be written as \[ f(x+iy) = \sum_{n=1}^\infty a_n\sqrt{y}K_{iR}(2\pi ny)\sin(2\pi nx). \] In both cases the real numbers $a_n$ are the coefficients of $f$ and are used to define its L-function $L(f,s)=\sum_{n=1}^\infty a_nn^{-s}$.