Level 8

I. Moonshine functions: Define, $$a(\tau)=\left(\frac{d_4^3}{d_2\,  d_8^2}\right)^{4},\quad b(\tau) = \left(\frac{d_1^2\,d_4}{d_2\,d_8^2}\right)^2, \quad c(\tau) = \left(\frac{d_2^5}{d_1^2\,d_4\,d_8^2}\right)^2$$  then, $$ \left(\frac{d_4^3}{d_2\,  d_8^2}\right)^{4}-4 = \left(\frac{d_1^2\,d_4}{d_2\,d_8^2}\right)^2\quad\\ \left(\frac{d_4^3}{d_2\,  d_8^2}\right)^{4}+4=\left(\frac{d_2^5}{d_1^2\,d_4\,d_8^2}\right)^2$$so \(a-4 = b,\, a+4 = c\). This \(a,b,c\) also belong to the class where \(a(\tau) = -a\big(\tfrac12+\tau\big)\) and \(b(\tau) = -c\big(\tfrac12+\tau\big)\). We then have the six level 8 moonshine functions,
$$\qquad\begin{align}
j_{8A}(\tau) &= \left(\frac{d_2\, d_4}{d_1\,  d_8}\right)^{8}\;=\;\frac{ac}b\\
j_{8B}(\tau) &= \left(\frac{d_4^2}{d_2\,  d_8}\right)^{12}=\sqrt{j_{4A}(2\tau)}\\
j_{8C}(\tau) &= \sqrt{j_{4B}(\tau)}\,=\,\sqrt[4]{j_{2A}(2\tau)}\\
&= \sqrt{j_{8A}(\tau)}-\frac{4}{\sqrt{j_{8A}(\tau)}}\\
j_{8D}(\tau) &= \left(\frac{d_2}{d_8}\right)^{4}\\
\color{red}{j_{8E}}(\tau) &= \left(\frac{d_4^3}{d_2\,  d_8^2}\right)^{4}\;=\; a\\
j_{8F}(\tau) &= \left(\frac{d_4}{d_8}\right)^{6}
\end{align}$$ Example: \(\quad j_{8C}\big(\tfrac14\sqrt{-58}\big) = 396\)

II. Relations: Using S1, this \(a,b,c\) explains all of the four trinomial identities of level 8 (prefixed with t8) by Somos. They also obey the relation,
$$\left(\frac{bc}{a}+64\frac{a}{bc}\right)+2a = \frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\qquad$$ In general, $$\left(\frac{bc}{a}+(b-c)^2\frac{a}{bc}\right)+2a = \frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\qquad$$ where \(b-c=8\). Or in terms of named functions of level 4 and 8,
$$\quad\quad j_{4B}\;+\;2\color{red}{j_{8E}} = j_{4D}+j_{8A}+j_{8A'}$$ where, $$j_{8A'}(\tau) = -j_{8A}\big(\tfrac12+\tau\big) = \left(\frac{d_1\, d_4^2}{d_2^2\,  d_8}\right)^{8}$$