of even weight given by Gundlach [@Gu]. This can be used to compute the Fourier expansions explicitly, which in turn can be employed to compute some CM values (mainly) numerically. For instance, for the CM point $z_0=(\zeta_5,\zeta_5^2)$ (where $\zeta_5=e^{2\pi i/5}$) corresponding to the cyclic CM extension $K={\mathbb{Q}}(\zeta_5)$ of $F={\mathbb{Q}}(\sqrt{5})$ we find that $$\begin{aligned}
\tilde \Psi_1(z_0) &=\frac{156973921227+70200871784\sqrt{5}}{156973921227-70200871784\sqrt{5}}=
\frac{\omega^{27}\cdot (4+\omega')^5\cdot(5+\omega')^5}{\omega'{}^{27}\cdot(4+\omega)^5\cdot(5+\omega)^5},\end{aligned}$$ where $\omega=\frac{1+\sqrt{5}}{2}$. By means of the results of [@BY] we derive a heuristic how the CM values of twisted Borcherds products should look like. In particular, we obtain a conjecture on the prime ideals ${\mathfrak l}\subset{\mathcal{O}}_F$ at which the CM value of a twisted Borcherds product can have non-zero order (see Conjecture \[conj\]). The same phenomenon as in [@GZ] and [@BY] should happen: Such prime ideals should be of small norm.

Finally, in Section \[sect:7\], we list some open problems for further research. For instance, in all the examples we computed it turned out that $\tilde \Psi_1({\mathcal{CM}}(K, \Phi, {\mathcal O}_F))$ belongs to the field $F$. Moreover, if $K/{\mathbb{Q}}$ is non-Galois then $\tilde
\Psi_1({\mathcal{CM}}(K, \Phi, {\mathcal O}_F))$ is square-free. According to Corollary \[clcor1.3\], this implies that for $z\in {\mathcal{CM}}(K, \Phi,
{\mathcal O}_F)$ the CM value $\tilde \Psi_1(z)$ generates the class field $L_K$ over 