In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as $\mathbb{R}^{d-1}\times\mathbb{R}^+$ has a Fourier series form: $$\phi(x,y)=y^\frac{d-1}{2}\sum_{n\in S\setminus 0}\rho(n)K_{it}(2\pi|n|y)e(\langle n, x\rangle),$$where $S\cong\mathbb{Z}^{d-1}$ is a lattice, $K_{it}$ is $K$-Bessel function, $e(z)=e^{2\pi iz}$ and $\langle,\rangle$ is usual inner product in $\mathbb{R}^{d-1}.$ Is there something known about the distribution of coefficients $\rho(n)$s?
More precisely I am interested in the following question. Let $X_{(t,N)}$ be the random variable which takes values $\rho(n)$ with $|n|^2=N$. Note that there are $\asymp N^{\frac{d-3}{2}}$ with $|n|^2=N$ (for $d\ge 3$). I have an intuition that $$\mathrm{Var}(X_{(t,N)})\ll t^c, \text{ for some }c<0.$$In other language for high eigenstates $n$'th Fourier coefficients are almost same for fixed norm of $n$. It is not very difficult to prove that $$\mathrm{Var}(X_{(t,N)})\ll N^pt^c \text{ for some }p>0,c<0.$$Also $\mathrm{Var}(X_{(t,1)})= 0$ trivially. I am trying to see why the variance is independent of $N$.
Thanks for any help and reference.
