Periodic Table of BKM Lie superalgebras

We list below the list of BKM Lie superalgebras that appear from counting $\mathbb{Z}_M$-twisted dyons in the CHL $\mathbb{Z}_N$-orbifold. The entry indicates the Siegel modular form and the associated cycle shape. The Siegel modular form provides the Weyl-Kac-Borcherds denominator identity of the BKM Lie superalgebra through its product and sum representations. The cycle shape leads to the Jacobi form that is the additive seed for the Siegel modular form that is the square of the modular form that appears in the table.

$\mathbf{N} \downarrow \Big\mid \mathbf{M} \rightarrow$ 1 2 3 4 5 6
1 $\Delta_5 / 1^{24}$ $\nabla_3 / 1^82^8$ $\nabla_2 / 1^63^6$ $\nabla_{3/2} / 1^42^2 4^4$ $\nabla_1 / 1^45^4$ $P_1 / 1^22^23^36^2$
2 $\widetilde{\nabla}_3 / 1^82^8$ $\Delta_2 / 2^{12}$ $Q_1 / 2^44^4$ ??
3 $\widetilde{\nabla}_2 /1^63^6$ $\Delta_1 / 3^{8}$
4 $\widetilde{\nabla}_{3/2} / 1^42^2 4^4$ $\widetilde{Q}_1 / 2^44^4$ $\Delta_{1/2} / 4^{6}$
5 ??
6 $\widetilde{P}_1 / 1^22^23^36^2$ ?? ??