The $L^{2}$-norm of Maass wave functions

Abstract:

Let $D$ denote the fundamental domain for the full modular group. Suppose that $f \in {L^2}(D)$ satisfies the wave equation $\Delta f = \lambda f$, where $\Delta$ is the noneuclidean Laplacian, and further, assume that $f$ is a common eigenfunction for all the Hecke operators. Then upper and lower bounds for the ${L^2}$-norm of $f$ are determined which depend only on $\lambda$ and the first Fourier coefficient of $f$.