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.