The Gaussian moments conjecture for Hecke–Maass cusp forms

Let $\mathbb{X}=\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, let $\{f_i\}$ be Hecke–Maass cusp forms normalized by

$$\frac{1}{\operatorname{vol}(\mathbb{X})}\int_{\mathbb{X}}|f_i(z)|^2\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}=1,$$

and define the Gaussian moments by $C_n=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}x^ne^{-x^2/2}\,\mathrm{d}x$, so that $C_n=(n-1)!!$ for even $n$ and $C_n=0$ for odd $n$. The Gaussian moments conjecture. For every $\psi\in C_c^\infty(\mathbb{X})$,

$$\int_{\mathbb{X}}\psi(z)f_i^n(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}\sim C_n\int_{\mathbb{X}}\psi(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}$$

as $t_{f_i}$ tends to infinity. This is the random-wave prediction that high-energy eigenfunctions have Gaussian value distributions; the source cites Berry and Humphries for this formulation.