  In his famous 2007 paper on three dimensional quantum gravity, Witten
  defined candidates for the partition functions
$$Z_k(q)=\sum_{n=-k}^{\infty}w_k(n)q^n$$ of potential extremal CFTs with
central charges of the form $c=24k$. Although such CFTs remain elusive, he
proved that these modular functions are well-defined. In this note, we point
out several explicit representations of these functions. These involve the
partition function $p(n)$, Faber polynomials, traces of singular moduli, and
Rademacher sums. Furthermore, for each prime $p\leq 11$, the $p$ series
$Z_k(q)$, where $k\in \{1, \dots, p-1\} \cup \{p+1\},$ possess a Ramanujan
congruence. More precisely, for every non-zero integer $n$ we have that $$
  w_k(pn) \equiv 0\begin{cases} \pmod{2^{11}}\ \ \ \ &{\text {\rm if}}\ p=2,
  \pmod{3^5} \ \ \ \ &{\text {\rm if}}\ p=3,
  \pmod{5^2}\ \ \ \ &{\text {\rm if}}\ p=5,
  \pmod{p} \ \ \ \ &{\text {\rm if}}\ p=7, 11.
  \end{cases}
  $$
