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 $$\begin{align}
\left(\frac{a}{b}\right)^2\left(\frac{c}{d}\right) &= \left(\frac{d_7\,d_9}{d_1\,d_{63}}\right)^3\\
\left(\frac{a}{b}\right)\left(\frac{c}{d}\right)^2 &= \left(\frac{d_{14}\,d_{18}}{d_2\,d_{126}}\right)^3\end{align}$$
Wednesday, November 6, 2019
Entry 64
Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) then for Level 42 $$\begin{align}
a &= \left(\frac{d_1^2\,d_{14}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right), \quad b = \left(\frac{d_2^2\,d_{7}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right)\\
c &= \left( \frac{d_3^2\,d_{42}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right), \quad d = \left(\frac{d_6^2\,d_{21}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right)\end{align}$$ The quadruple \((a,b,c,d)\) obey $$\begin{align}a\, b\, c\, d &=1\\ a+3 &= b\\ c+1 &= d\end{align}$$ and their ratios are squares$$\begin{align}\frac{a}{d} &= \left(\frac{d_1\,d_{14}}{d_6\,d_{21}}\right)^2\\ \frac{b}{c} &= \left(\frac{d_2\,d_{7}}{d_3\,d_{42}}\right)^2\end{align}$$
a &= \left(\frac{d_1^2\,d_{14}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right), \quad b = \left(\frac{d_2^2\,d_{7}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right)\\
c &= \left( \frac{d_3^2\,d_{42}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right), \quad d = \left(\frac{d_6^2\,d_{21}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right)\end{align}$$ The quadruple \((a,b,c,d)\) obey $$\begin{align}a\, b\, c\, d &=1\\ a+3 &= b\\ c+1 &= d\end{align}$$ and their ratios are squares$$\begin{align}\frac{a}{d} &= \left(\frac{d_1\,d_{14}}{d_6\,d_{21}}\right)^2\\ \frac{b}{c} &= \left(\frac{d_2\,d_{7}}{d_3\,d_{42}}\right)^2\end{align}$$
Entry 63
For certain even levels divisible by \(7\) such as \(7\times4,\, 7\times6,\, 7\times18\) or \(28, 42, 126\), we may still find an eta quotient \(a\) such that \(a+m = b\) is also an eta quotient, but only for one integer \(m\). Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) and $$a=\left(\frac{d_1\,d_7}{d_4\,d_{28}}\right),\quad b = \left(\frac{d_2^3\,d_{14}^3}{d_1\,d_4^2\,d_7\,d_{28}^2}\right)$$ then $$\begin{align}a+2 &= b\\ a+\frac{4}{a}+4 & = \left(\frac{d_2^2\,d_{14}^2}{d_1\,d_4\,d_7\,d_{28}}\right)^3\\ b+\frac{4}{b}-4 &= \left(\frac{d_1\,d_7}{d_2\,d_{14}}\right)^3\end{align}$$ where \((a,b)\) is the McKay-Thompson series of class 28C for the Monster (A161970).
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