Entry 41

Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then for Level 6 $$\left(\frac{d_2^2\,d_3}{d_1\,d_6^2}\right)^4-1=\left(\frac{d_2\,d_3^3}{d_1\,d_6^3}\right)^3\\ \left(\frac{d_2^2\,d_3}{d_1\,d_6^2}\right)^4-9=\left(\frac{d_1^5\,d_3}{d_2\,d_6^5}\right)$$ Expressed as the triple of eta quotients \((a,b,c)\) such that $$a-1 =b\\ a-9=c$$ then \((m,n)=(-1,-9)\) and $$ \frac{ab}{c} =\left(\frac{d_2\,d_3}{d_1\,d_6}\right)^{12}\\ \frac{bc}{a} = \left(\frac{d_1\,d_3}{d_2\,d_6}\right)^6\\ \frac{ac}{b} = \left(\frac{d_1\,d_2}{d_3\,d_6}\right)^4$$ where \((a,b,c)\) are the McKay-Thompson series of class 6E (A105559, A128633). They obey $$\left(\frac{64a}{bc}+\frac{bc}a\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b}$$ which is one of the \(9\) dependencies found by Conway, Norton, and Atkin such that the moonshine functions span a linear space of \(172-9=163\) dimensions. Similar identities involving only moonshine functions exist for levels \((6, 10, 12, 18, 30)\).