$M$. It is an interesting question whether this is a general phenomenon.

We mention that J. Rouse has used Theorem 1.1 to determine the Fourier coefficients of modular functions $f \in \mathbb{Z}[j]$ in terms of traces of singular moduli [@Rouse].

We thank Heike Hippauf and Sebastian Mayer for their valuable help with the computations in Section \[sect:examples\]. Moreover, we thank Lev Borisov, Bas Edixhoven, and Don Zagier for interesting and useful discussions, and thank Shou-Wu Zhang for bringing to our attention his equidistribution theorem, which is needed in the proof of Theorem \[cltheo6.6\].

Non-holomorphic Poincaré series {#sect:2}
===============================

Here we consider non-holomorphic Poincare series of weight $0$. The results of this section are known. We state them for completeness and to fix the notation. For details we refer to [@He], [@Ni], [@Br2].

Let $I_\nu(z)$ and $K_\nu(z)$ be the usual modified Bessel functions as in [@AS] §10. For convenience we put for $s\in{\mathbb{C}}$ and $y\in{\mathbb{R}}\setminus \{0\}$: $$\begin{aligned}
\label{calI}
{\mathcal{I}}_s(y)&=  \sqrt{\frac{\pi |y|}{2}}I_{s-1/2}(|y|),\\
{\mathcal{K}}_s(y)&= \label{calK}
\sqrt{\frac{2|y|}{\pi}}K_{s-1/2}(|y|).\end{aligned}$$ The functions ${\mathcal{I}}_s(y)$ and ${\mathcal{K}}_s(y)$ are holomorphic in $s$. At $s=1$ they have the special values $$\begin{aligned}
\label{Mspecial}
{\mathcal{I}}_{1}(y)&=\sinh(|y|), \\
\label{Wspecial}
{\mathcal{K}}_{1}(y)&=e^{-|y|},\\
\label{MWspecial} 2{\mathcal{I}}_{1}(y)+{\mathcal{K}}_{1}(y)&=e^{|y|}.\end{aligned}$$

The full elliptic modular group $\Gamma'={\operatorname{SL}}_2({\mathbb{Z}})$ acts on the upper complex half plane 