Levels 126 & 252

I. Non-moonshine functions. There are no moonshine functions with uppercase index (in Atlas notation) for level \(>119\). However, surprisingly we can still find trinomial identities for level 144 (some a consequence of level 72) and as high as level 252 (a consequence of level 126). For the latter, define the two pairs of eta quotients, $$\begin{align}
a(\tau) &= \left(\frac{d_7^2\,d_{9}^2}{d_3\,d_{14}\,d_{18}\,d_{21}}\right), \quad b(\tau) = \left(\frac{d_1^2\,d_{63}^2}{d_2\,d_3\,d_{21}\,d_{126}}\right)\\
c(\tau) &= \left( \frac{d_{14}^2\,d_{18}^2}{d_6\,d_7\,d_{9}\,d_{42}}\right),\;\quad d(\tau) = \left(\frac{d_2^2\,d_{126}^2}{d_1\,d_6\,d_{42}\,d_{63}}\right)\end{align}$$Given one moonshine function of level 21, namely \(j_{21B}(\tau) = \left(\frac{d_1\,d_3}{d_7\,d_{21}}\right)\) and define, $$e(\tau) = \frac{j_{21B}(\tau)}{j_{21B}(3\tau)} = \left(\frac{d_1\,d_{63}}{d_7\,d_9}\right)$$ then ratios of the pairs are simply, $$\begin{align}
\frac{a}{b} &= \frac{e(2\tau)}{e^2(\tau)}\\
\frac{c}{d} &= \frac{e(\tau)}{e^2(2\tau)}\\\
\end{align}$$They obey,$$\begin{align}
a(\tau)-2 &= b(\tau)\\
c(\tau)-1 &= d(\tau)\end{align}$$ $$\begin{align}
a\big(\tfrac12+\tau\big)-2 &= b\big(\tfrac12+\tau\big)\\
c\big(\tfrac12+\tau\big)-1 &= d\big(\tfrac12+\tau\big)\end{align}$$ which is the pair of trinomial identities each for level 126 and level 252, respectively, and the latter seems to be the highest known.