Entry 69

II. Level 2. The McKay-Thompson series of class 2B for the Monster.

$$j_{2B}(\tau) =\left(\frac{d_1}{d_2}\right)^{24}$$ Examples: 

$$\begin{align}j_{2B}\big(\tfrac12\sqrt{-10}\big) & =\, 2^6\,U_{5}^{12} = 2^6\left(\tfrac{1+\sqrt{5}}2\right)^{12}\\ j_{2B}\big(\tfrac12\sqrt{-58}\big) & = \,2^6\,U_{29}^{12} = 2^6\left(\tfrac{5+\sqrt{29}}2\right)^{12}\\  j_{2B}\Big(\tfrac{1+\sqrt{-5}}2\Big) & = -2^6\,U_{5}^6 = -2^6\left(\tfrac{1+\sqrt{5}}2\right)^6\\ j_{2B}\Big(\tfrac{1+\sqrt{-13}}2\Big) & = -2^6\,U_{13}^6 = -2^6\left(\tfrac{3+\sqrt{13}}2\right)^6\\ j_{2B}\Big(\tfrac{1+\sqrt{-37}}2\Big) & = -2^6\,U_{37}^6 = -2^6\big(6+\sqrt{37}\big)^6\end{align}$$ which have class number \(h(-d)=2\).  The quadratic irrationals \(U_n\) are fundamental units, solutions to Pell equations \(x^2-ny^2=-1\) as discussed in Level 1.