Entry 65

For level \(>119\), there are no more moonshine functions with uppercase index (in Atlas notation). Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) then for Level 126, $$\begin{align}
a &= \left(\frac{d_7^2\,d_{9}^2}{d_3\,d_{14}\,d_{18}\,d_{21}}\right), \quad b = \left(\frac{d_1^2\,d_{63}^2}{d_2\,d_3\,d_{21}\,d_{126}}\right)\\
c &= \left( \frac{d_{14}^2\,d_{18}^2}{d_6\,d_7\,d_{9}\,d_{42}}\right),\;\;\quad d = \left(\frac{d_2^2\,d_{126}^2}{d_1\,d_6\,d_{42}\,d_{63}}\right) \end{align}$$ The quadruple \((a,b,c,d)\) obey $$\begin{align}
a-2 &= b\\
c-1 &= d\end{align}$$ and their ratios are cubes $$\begin{align}
\left(\frac{a}{b}\right)^2\left(\frac{c}{d}\right) &= \left(\frac{d_7\,d_9}{d_1\,d_{63}}\right)^3\\
\left(\frac{a}{b}\right)\left(\frac{c}{d}\right)^2  &=  \left(\frac{d_{14}\,d_{18}}{d_2\,d_{126}}\right)^3\end{align}$$