Entry 53

Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then for Level 30

$$\begin{align}\frac{d_6\, d_{10}}{d_3\, d_5}\left(\frac{d_1\, d_{15}}{d_2\, d_{30}}\right)^2+1 &= \frac{d_1\, d_{15}}{d_3\, d_5}\left(\frac{d_6\, d_{10}}{d_2\, d_{30}}\right)^2\\ \frac{d_6\, d_{10}}{d_3\, d_5}\left(\frac{d_1\, d_{15}}{d_2\, d_{30}}\right)^2+2 &= \left(\frac{d_3\, d_5}{d_2\, d_{30}}\right)\end{align}$$

Expressed as the triple of eta quotients \((a,b,c)\) such that $$a+1 =b\\ a+2=c$$ then \((m,n)=(1,2)\) and
$$\begin{align}
\frac{ab}c &= \left(\frac{d_1\, d_6\, d_{10}\, d_{15}}{d_2\,  d_{3}\, d_{5}\, d_{30}}\right)^3\\ \frac{bc}a &= \left(\frac{d_3\, d_5\, d_{6}\, d_{10}}{d_1\,  d_{2}\, d_{15}\, d_{30}}\right)\\ \frac{ac}b &= \left(\frac{d_1\, d_3\, d_{5}\, d_{15}}{d_2\,  d_{6}\, d_{10}\, d_{30}}\right)\end{align}$$ where \((a,b,c)\) are the McKay-Thompson series of class 30G (A133098). They obey $$\left(\frac{a}{bc}+\frac{bc}a\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b}$$ 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 \((6, 10, 12, 18, 30)\). So this is the highest level.