Level 12, part 1

Level 12 is very special because, given an eta quotient $a$, we can find $a+m_i$ as an eta quotient for four different integer values $m_i$. However, that also makes the situation more complicated than in other levels. Let,
$$a(\tau) = \left(\frac{d_4^2\,d_6}{d_2\,d_{12}^2}\right)^2,\quad b(\tau) = \frac{d_3^3\,d_4}{d_1\,d_{12}^3}, \quad c(\tau) =\frac{d_1^3\,d_4\, d_6^2}{d_2^2\,d_3\,d_{12}^3}$$ such that $a+1 = b,\,a-3 = c.$ And the monster functions, $$\begin{align} j_{12A}(\tau) &= \left(\frac{d_2^2\, d_6^2}{d_1\,d_3\,d_4\, d_{12}}\right)^{6}\\ \color{red}{j_{12B}}(\tau) &= \left(\frac{d_1\, d_4\,d_6}{d_2\,d_3\,  d_{12}}\right)^4\,=\,\frac{ac}b\\ j_{12C}(\tau) &= \sqrt{j_{6A}(2\tau)}\\ &=j_{12F}(\tau)+\frac{1}{j_{12F}(\tau)}\\ &= j_{12G}(\tau)+\frac{9}{j_{12G}(\tau)}\\ j_{12D}(\tau) &= \left(\frac{d_6^2}{d_3\,  d_{12}}\right)^{8}=\sqrt[3]{j_{4A}(3\tau)}\\ \color{blue}{j_{12E}}(\tau) &= \left(\frac{d_1\, d_3}{d_4\,  d_{12}}\right)^{2}\;=\;\;\frac{bc}a\\ j_{12F}(\tau) &= \left(\frac{d_4\, d_6}{d_2\,  d_{12}}\right)^{6}=\sqrt{j_{6B}(2\tau)}\\ j_{12G}(\tau) &= \left(\frac{d_2\, d_4}{d_6\,  d_{12}}\right)^{2}=\sqrt{j_{6D}(2\tau)}\\ \color{blue}{j_{12H}}(\tau) &= \left(\frac{d_3\, d_4}{d_1\,  d_{12}}\right)^{4}\;=\;\frac{ab}c\\ \color{red}{j_{12I}}(\tau) &= \left(\frac{d_4^2\, d_6}{d_2\,  d_{12}^2}\right)^{2}\;=\; a\\ j_{12J}(\tau) &= \;\left(\frac{d_6}{d_{12}}\right)^{4}\; = \;\sqrt{j_{6F}(2\tau)} \end{align}$$
Example: $\quad j_{12C}\left(\tfrac12\sqrt{-17/6}\right) = 198$

Let $a=\color{red}{j_{12I}},\;m = 1,\;n=-3,$ then,
$$\begin{align} j_{12E} &= a-\frac{3}{a}-2\\ j_{12B} &= a+1+\frac{4}{a+1}-5\\ j_{12H} &= a-3+\frac{12}{a-3}+7 \end{align}$$ as well as,
$$j_{12A} =  j_{12E}+\frac{4^2}{j_{12E}}+8 =  j_{12B}+\frac{3^2}{j_{12B}}+10 =  j_{12H}+\frac{1}{j_{12H}}+2$$ which implies the linear relation between 5 monster functions,
$$j_{12A}+2\color{red}{j_{12I}} =\color{red}{ j_{12B}}+\color{blue}{j_{12E}}+\color{blue}{j_{12H}}+8$$ The functions $j_{12I}$ and  $j_{12B}$ can be used for level 12, while versions of $j_{12E}$ and  $j_{12H}$ will be useful in level 24.