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).