The traces-of-singular-moduli conjecture for Thompson McKay–Thompson series

Let $g\in \textsl{Th}$, with $g$ not in the classes $\mathrm{9B}$, $\mathrm{18B}$ or $\mathrm{27ABC}$. Let $\breve{F}_{\mathrm{3C},g}(\tau)$ be the McKay–Thompson series, let $q=e^{2\pi i\tau}$, let $\theta$ be the theta function, let $\hat{\Gamma}_g$ and $\chi_g$ be the auxiliary group and generalized genus character defined above, and let $\operatorname{Tr}_{-3,\hat{\Gamma}_g}(\widetilde{T}_{\mathrm{3C},g}^3,D,\chi_g)$ denote the corresponding trace. Define coefficients $\kappa_{g,n^2}$ by requiring that they all vanish except $\kappa_{\mathrm{9A},9}=-9$, $\kappa_{\mathrm{18A},9}=-1$, $\kappa_{\mathrm{36ABC},9}=-3$, and $\kappa_{\mathrm{36ABC},36}=4$. For the class $\mathrm{9B}$, use the displayed quadratic-form trace in the claim below. Traces-of-singular-moduli conjecture. For $g\in\textsl{Th}$ not in the classes $\mathrm{9B}$, $\mathrm{18B}$ or $\mathrm{27ABC}$, one has

$$\breve{F}_{\mathrm{3C},g}(\tau)=3q^{-3}-3\operatorname{tr}(g|\boldsymbol{248})\frac{\theta(\tau)-1}{2}+3\sum_{n>1}\kappa_{g,n^2}\frac{\theta(n^2\tau)-1}{2}+3\sum_{D>0}\frac{1}{\sqrt D}\operatorname{Tr}_{-3,\hat{\Gamma}_g}(\widetilde{T}_{\mathrm{3C},g}^3,D,\chi_g)q^D.$$

For $g$ in the class $\mathrm{9B}$, one has

$$\breve{F}_{\mathrm{3C},\mathrm{9B}}(\tau)=3q^{-3}-3\operatorname{tr}(g|\boldsymbol{248})\frac{\theta(\tau)-1}{2}+3\sum_{D>0}\frac{1}{\sqrt D}\left(\sum_{Q\in\mathcal{Q}^{(9)}_{-3D}/\Gamma_0(3)}\chi_{-3}(Q)\frac{\widetilde{T}_{\mathrm{3C},g}(\alpha_Q)^3}{|\overline{\Gamma_0(3)}_Q|}\right)q^D.$$

These formulas propose a systematic realization of the inverse image of generalized monstrous moonshine under singular theta lifts, expressing the relevant Thompson McKay–Thompson series through traces of the principal moduli $\widetilde{T}_{\mathrm{3C},g}^3$; their status is not resolved in the supplied source.