Levels 42 & 84

I. Moonshine functions. The four functions for level 42,
$$\begin{align}
j_{42A}(\tau) &= \left(\sqrt{j_{42B}}+\frac1{\sqrt{j_{42B}}}\right)^2\\
j_{42B}(\tau) &= \left(\frac{d_1\,d_6\,d_{14}\,d_{21}}{d_2\,d_3\,d_7\,d_{42}}\right)^2\\
j_{42C}(\tau) &= \sqrt[3]{j_{14B}(3\tau)} = \left(\frac{d_3\,d_{21}}{d_6\,d_{42}} \right)\\
j_{42D}(\tau) &= \left(\frac{d_2\,d_6\,d_7\,d_{21}}{d_1\,d_3\,d_{14}\,d_{42}} \right)
\end{align}$$ and the three functions for level 84,
$$\begin{align}
j_{84A}(\tau) &= \sqrt{j_{42A}(2\tau)}\\
j_{84B}(\tau) &= \sqrt{j_{42B}(2\tau)}\\
j_{84C}(\tau) &= \sqrt[3]{j_{28B}(3\tau)}= \left(\frac{d_6^2\,d_{42}^2}{d_3\,d_{12}\,d_{21}\,d_{84}}\right)\\
\end{align}$$
II. Non-moonshine functions. Define the two pairs of eta quotients, $$\begin{align}
a(\tau) &= \left(\frac{d_1^2\,d_{14}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right), \quad b(\tau) = \left(\frac{d_2^2\,d_{7}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right)\\
c(\tau) &= \left( \frac{d_3^2\,d_{42}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right), \quad d(\tau) = \left(\frac{d_6^2\,d_{21}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right)\end{align}$$ Interestingly, these satisfy $$a(\tau)\, b(\tau)\, c(\tau)\, d(\tau) = 1$$ And in terms of the moonshine functions,
$$\begin{align}
\frac{a}{b} &= \frac{j_{42B}}{j_{42D}}\\
\frac{d}{c} &= j_{42B}\,j_{42D}\\
\end{align}$$ They obey, $$\begin{align}
a(\tau)+3 &= b(\tau)\\
c(\tau)+1 &= d(\tau)\end{align}$$ $$\begin{align}
a\big(\tfrac12+\tau\big)+3 &= b\big(\tfrac12+\tau\big)\\
c\big(\tfrac12+\tau\big)+1 &= d\big(\tfrac12+\tau\big)\end{align}$$ which is the pair of trinomial identities each for level 42 and level 84, respectively.