Entry 50

Level 24, Part 1. We can find multiple triples \((a,b,c)\), but Level 24 is unusual since two have the special property $$\frac{a_1}{b_1\, c_1} = \frac{a_2}{b_2\, c_2}$$ Naturally, these also form linear dependencies but involve non-monster functions. However, if we combine the two, plus some level 12 functions, then there is a relationship with only monster functions. Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then $$\qquad \left(\frac{d_6\, d_{12} }{d_2\, d_ 4}\right) \left(\frac{d_1\, d_8 }{d_3\, d_{24} }\right)^2+1=\left(\frac{d_1\, d_8 }{d_2\, d_4 }\right) \left(\frac{d_6\, d_ {12}}{d_3\, d_{24} }\right)^3\\ \left(\frac{d_6\, d_{12} }{d_2\, d_ 4}\right) \left(\frac{d_1\, d_8 }{d_3\, d_{24} }\right)^2+3= \frac{\big( d_2\, d_4\big)^2 }{d_1\, d_3\, d_8\, d_{24} }\qquad $$ Expressed as the triple of eta quotients \((a,b,c)\) such that $$a+1 =b\\ a+3=c$$ so \((m,n)=(1,3)\). Then $$\begin{align}\frac{ab}{c} &= \left(\frac{d_1\,d_6\,d_8\,d_{12}}{d_2\,d_3\,d_4\,d_{24}}\right)^4\\ \color{blue}{\frac{bc}{a}} &= \left(\frac{d_2\, d_4\, d_6\, d_{12}}{d_1\, d_3\, d_8\, d_{24}}\right)^2 = T_{24B}\\ \frac{ac}{b} &=  \left(\frac{d_2\, d_4}{d_6\,d_{12}}\right)^2 \end{align}$$ where \((a,b,c)\) are the McKay-Thompson series of class 24C (A206298). They obey $$\frac{bc}a = a+\frac3a+4$$ $$\left(\frac{4a}{bc}+\frac{bc}{a}\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b} $$ However, this is not one of the 9 linear dependencies by Conway et al since one of the terms, namely \(\left(\frac{ab}c\right)\), is not a moonshine function. But the \(\left(\frac{bc}a\right)\) term is the McKay-Thompson series of class 24B (A212771) and also appears in Part 2.