Entry 71

III. Level 3. The McKay-Thompson series of class 3B for the Monster.

$$j_{3B}(\tau) =\left(\frac{d_1}{d_3}\right)^{12}$$ Examples: $$\begin{align}j_{3B}\Big(\tfrac{1+\sqrt{-5/3}}{2}\Big) &= -3^3U_{5}^2 =-3^3\left(\tfrac{1+\sqrt{5}}2\right)^2\\ j_{3B}\Big(\tfrac{1+\sqrt{-17/3}}{2}\Big) &= -3^3U_{17}^2 =-3^3\left(4+\sqrt{17}\right)^2\\ j_{3B}\Big(\tfrac{1+\sqrt{-41/3}}{2}\Big) &=-3^3U_{41}^2 =-3^3\left(32+5\sqrt{41}\right)^2\\ j_{3B}\Big(\tfrac{1+\sqrt{-89/3}}{2}\Big) &= -3^3U_{89}^2=-3^3\left(500+53\sqrt{89}\right)^2\end{align}$$ These \(d=3m\) have class number \(h(-d)=2\). The quadratic irrationals have already appeared in Level 1 and are fundamental units \(U_n\), solutions to Pell equations \(x^2-ny^2=\pm1\). See also class 7B.