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 $\eta(\tau)$.


  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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.