Entry 82

Level 13. Define $d_k = \eta(k\tau)$ with Dedekind eta function $\eta(k\tau)$ and the McKay-Thompson series of class 13A for the Monster. $$j_{13A}(\tau) = \left(\frac{d_1}{d_{13}}\right)^2+13\left(\frac{d_{13}}{d_1}\right)^2+6$$

Examples. We select $d=13m$ for class number $h(-d)=2$ with $m=7, 31$ as well as class number $h(-d)=6$ with $m=19, 151$ and find a well-behaved pattern just like for level 5 $$\begin{align}j_{13A}\Big(\tfrac{1+\sqrt{-7/13}}{2}\Big) & = -(\sqrt7)^2\\ j_{13A}\Big(\tfrac{1+\sqrt{-31/13}}{2}\Big) & = -(2\sqrt{31})^2\\ j_{13A}\Big(\tfrac{1+\sqrt{-19/13}}{2}\Big) & = -(x\sqrt{19})^2\\ j_{13A}\Big(\tfrac{1+\sqrt{-151/13}}{2}\Big) & = -(y\sqrt{151})^2\end{align}$$

where $(x,y)$ are roots of cubics $$19x^3 - 38x^2 + 21x - 9 = 0\\ 151y^3 - 2567y^2 - 512y - 44 = 0$$ and so on for class number $h(-d)=10$, etc.