# Monstrous moonshine for $M_{24}$ and K3?

An important piece of Monstrous moonshine is the j-function,

$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$

In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and Harvey define the function,

$$H^{(2)}(\tau)=2q^{-1/8}(-1 + 45q + \color{blue}{231}q^2 + 770q^3 + 2277q^4 + 5796q^5+\dots)\tag{2}$$

It was first observed in Notes on the K3 Surface and the Mathieu group M24 (2010) by Eguchi, Ooguri, and Tachikawa that the first five coefficients of $(2)$ are equal to the dimensions of irreducible representations of $M_{24}$.

Edit (Nov. 23)

For info, in the paper cited by J.Harvey below, in page 44, eqn(7.16) and (7.19), the authors missed a tiny but crucial + sign $n\in\mathbb{Z^{\color{red}{+}}}$ in the summation:

\begin{aligned}h^{(2)}(\tau)&=\frac{\vartheta_2(0,p)^4-\vartheta_4(0,p)^4}{\eta(\tau)^3}-\frac{24}{\vartheta_3(0,p)}\sum_{n\in\mathbb{Z^{\color{red}{+}}}}\frac{q^{n^2/2-1/8}}{1+q^{n-1/2}}\\ &=q^{-1/8}(-1+45q+231q^2+770q^3+2277q^4+\dots)\end{aligned}

where $q = p^2$, nome $p = e^{\pi i \tau}$, Jacobi theta functions $\vartheta_n(0,p)$, and Dedekind eta function $\eta(\tau)$.

Questions:

1. Does anyone know how to compute the rest of the coefficients of $(2)$? (The OEIS only has the first nine.)

2. For $\tau=\tfrac{1+\sqrt{-163}}{2}$, is $H^{(2)}(\tau)$ algebraic or transcendental? Does it have a neat closed-form expression like $j(\tau) = -640320^3$?

3. Is the appearance of $\color{blue}{231}$ in $j\big(\tau) = -12^3(231^2-1)^3$ a coincidence? (Note also that its smaller sibling $j\big(\tfrac{1+\sqrt{-67}}{2}\big) = -12^3(21^2-1)^3$ and $M_{23}$ has dimensions $1,42,210$.)

4. Likewise, in $\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12} = -3^5 \left(231-\sqrt{2(-26679+2413\sqrt{3\cdot163})} \right)^{-2}$ where $\eta(\tau)$ is the Dedekind eta function.

I can answer your first question. In arXiv:1208.4074 by Dabholkar, Murthy and Zagier you can find a formula that implies $H^{(2)}(\tau)= \frac{48 F_2^{(2)}(\tau)- 2 E_2(\tau)}{\eta(\tau)^3}$ where $E_2(\tau)$ is the quasi modular Eisenstein series and $F_2^{(2)}(\tau)= \sum_{r>s>0,r-s\ \mathrm{odd}} (-1)^r s\, q^{rs/2}$ which you can use to compute $H^{(2)}(\tau)$ to whatever order you desire. I have no idea concerning your questions 2,3, and 4.