Entry 66

Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\).

The McKay-Thompson series of class 1A for the Monster (A007240), disregarding the constant term \(744\), is the well-known j-function \(j(\tau)\). Given the three Weber modular functions \(\mathfrak{f}_n\) 

$$\begin{align}j_{1A}(\tau) &=\left(\frac{\mathfrak{f}(\tau)^{16}+\mathfrak{f}_1(\tau)^{16}+\mathfrak{f}_2(\tau)^{16}}{2}\right)^3\\ &= \left(\left(\frac{d_1}{d_2}\right)^8+2^8\left(\frac{d_2}{d_1}\right)^{16}\right)^3\end{align}$$ They tend to be cubes, but not always. Examples: 

$$\begin{align}j_{1A}\big(\tfrac{1+\sqrt{-67}}{2}\big) &= -12^3\big(21^2-1\big)^3 \,=\, -5280^3\\ j_{1A}\big(\tfrac{1+\sqrt{-163}}{2}\big) &= -12^3\big(231^2-1\big)^3=-640320^3\end{align}$$ while the two highest \(d\) with class number \(h(-d)=2\) are

$$\; j_{1A}\big(\tfrac{1+\sqrt{-403}}{2}\big)=-12^3\big((5301 + 1470\sqrt{13})^2-1\big)^3\\ j_{1A}\big(\tfrac{1+\sqrt{-427}}{2}\big)=-12^3\big((7215 + 924\sqrt{61})^2-1\big)^3$$ and so on, though if \(d\) is a multiple of \(3\) then it is almost a cube $$\begin{align}j_{1A}\big(\tfrac{1+\sqrt{-15}}2\big) &=-3^3\,U_5^2\,\big(5+4\sqrt{5}\big)^3\\ j_{1A}\big(\tfrac{1+\sqrt{-51}}2\big) &=-48^3\,U_{17}^2\,\big(5+\sqrt{17}\big)^3\\ j_{1A}\big(\tfrac{1+\sqrt{-123}}2\big) &=-480^3\,U_{41}^2\,\big(8+\sqrt{41}\big)^3\\ j_{1A}\big(\tfrac{1+\sqrt{-267}}2\big) &=-240^3\,U_{89}^2\,\big(625+53\sqrt{89}\big)^3\end{align}$$ where \(U_n\) are fundamental units, $$\begin{align}U_5 &= \tfrac{1+\sqrt{5}}2\\ U_{17} &= 4+\sqrt{17}\\ U_{41} &= 32+5\sqrt{41}\\ U_{89} &= 500+53\sqrt{89}\end{align}$$ or fundamental solutions to the Pell equation \(x^2-ny^2 = -1\) with the first as the golden ratio \(\phi\ =U_5\). These will appear later in Class 3B.