Levels 1, 2, 3

Recall our notation for the Dedekind eta function, $$\color{blue}{d_m = \eta(m\tau)}$$ where we suppress the argument \(\tau\) for ease of notation. Then we have the following moonshine functions:

Level 1
$$\begin{align}
j_{1A}(\tau) &= \left(\left(\frac{d_1}{d_2}\right)^8+2^8\left(\frac{d_2}{d_1}\right)^{16}\right)^3
\end{align}$$ This is the well-known j-function \(j(\tau)\). In this form, it is easily seen to be a cube. Its most famous value is \(j_{1A}\big(\tfrac{1+\sqrt{-163}}{2}\big)=-640320^3\) .

Level 2
$$\begin{align}
j_{2A}(\tau) &=\left(\left(\frac{d_1}{d_2}\right)^{12}+2^6\left(\frac{d_2}{d_1}\right)^{12}\right)^2\\
j_{2B}(\tau) &= \left(\frac{d_1}{d_2}\right)^{24}
\end{align}$$ Alternatively, $$j_{2A}(\tau) =\left(\left(\frac{d_1\,d_2}{d_{1/2}\;d_4}\right)^4-4\left(\frac{d_{1/2}\;d_4}{d_1\,d_2}\right)^4\right)^4$$ In this form, \(j_{2A}\) is seen to be a 4th power. The function doesn't have a name, but is notable for the value \(j_{2A}(\tfrac12\sqrt{-58})=396^4\) in Ramanujan's well-known pi formula. Note that,
$$j_{4B}(\tau) = \sqrt{j_{2A}(2\tau)}\\j_{8C}(\tau) = \sqrt[4]{j_{2A}(2\tau)}$$
Level 3
$$\begin{align}
j_{3A}(\tau) &= \left(\left(\frac{d_1}{d_3}\right)^6+3^3\left(\frac{d_3}{d_1}\right)^6\right)^2\\
j_{3B}(\tau) &=  \left(\frac{d_1}{d_3}\right)^{12}\\
j_{3C}(\tau) &=  \sqrt[3]{j_{1A}(3\tau)}
\end{align}$$ Alternatively, $$j_{3A}(\tau) = \left(\frac{d_1^2}{d_3^2}+\frac{9\,d_9^3}{d_1\,d_3^2}\right)^6$$ In this form, \(j_{3A}\) is seen to be also a cube. An example is \(j_{3A}\big(\tfrac{1+\sqrt{-89/3}}{2}\big)=-300^3\) which can also be used in a Ramanujan-type pi formula. Notice that \(j_{3C}\) is a cube root.