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).