Entry 72

 IV. Level 4. The McKay-Thompson series of class 4A for the Monster (A097340)

$$\begin{align}j_{4A}(\tau) &= \left(\left(\frac{d_1}{d_4}\right)^4+4^2\left(\frac{d_4}{d_1}\right)^4\right)^2\\ &=\left(\frac{d_2^2}{d_1\,d_4}\right)^{24}\end{align}$$ The second form shows they can be \(12\)th powers. Examples: 

$$j_{4A}\big(\tfrac12\sqrt{-7}\big)=2^{12}$$

which has class number 1 (but non-fundamental \(d\)). For class number 2,

$$\begin{align}j_{4A}\Big(\tfrac{1+\sqrt{-6}}{2}\Big) &= -2^6\left(1+\sqrt2\right)^{4}\\ j_{4A}\Big(\tfrac{1+\sqrt{-10}}{2}\Big) &= -2^6\left(\frac{1+\sqrt5}2\right)^{12}\\ j_{4A}\Big(\tfrac{1+\sqrt{-22}}{2}\Big) &= -2^6\left(1+\sqrt2\right)^{12}\\ j_{4A}\Big(\tfrac{1+\sqrt{-58}}{2}\Big) &= -2^6\left(\frac{5+\sqrt{29}}2\right)^{12}\end{align}$$