This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield,
$$\begin{aligned}h^{(2)}(\tau)&=\frac{\vartheta_2(0,p)^4-\vartheta_4(0,p)^4}{\eta(\tau)^3}-\frac{24}{\vartheta_3(0,p)}\sum_{n=-\infty}^\infty\frac{q^{n^2/2-1/8}}{1+q^{n-1/2}}\\ &=\color{red}{m}\,q^{-1/8}(-1+45q+231q^2+770q^3+2277q^4+\dots)\end{aligned}$$
It was observed by Eguchi, Ooguri, and Tachikawa that the first five coefficients of the RHS are equal to the dimensions of irreducible representations of $M_{24}$.
I assume that $q = p^2$, nome $p = e^{\pi i \tau}$, Jacobi theta functions $\vartheta_n(0,p)$, Dedekind eta function $\eta(\tau)$, and 30 coefficients $a_i$ of the RHS are given by OEIS A212301 as $2a_i$.
Question:
The paper implies that $m=1$. However, if I test it with $\tau=\sqrt{-n}$ for various positive integer $n$, then it seems m varies as well. In particular, if $\tau=\sqrt{-1}$, then apparently $m=2$. Which of my assumptions are wrong, and how do we fix it? (Or is it a bug in Mathematica again?)
Neither you nor Mathematica are to blame.
If you change the coefficient $24$ in front of the sum to $12,$ the expression produces the desired series. Alternatively, following ccorn's comment, you can change the sum so that it runs from $1$ to $\infty$ rather than $-\infty$ to $\infty$. In the paper you cite, the authors write $\displaystyle\sum_{n\in\mathbb Z}.$ Perhaps they meant $\displaystyle\sum_{n\in\mathbb Z^+}.$
(I observed this by expanding both terms in the expression and noting that, after removing the overall factor $p^{-1/4},$ both contain undesired odd powers of $p.$ These undesired terms, however, differ between the two terms only by a factor of $2,$ and will cancel if the second term is halved.)