The McKay–Thompson moonshine conjecture

Let $g$ be an element of the Monster group, and let $j_g(q)$ be its McKay–Thompson series. For a prime level $N$, let $\Gamma_0(N)$ be the congruence subgroup of the modular group and let $\Gamma_0(N)^+$ denote its normaliser in $SL(2;\mathbb{Z})$. Moonshine conjecture. The McKay–Thompson series $j_g(q)$ is the unique modular invariant of a group lying between $\Gamma_0(N)$ and its normaliser $\Gamma_0(N)^+$ in the modular group $SL(2;\mathbb{Z})$. This conjecture is part of monstrous moonshine, which relates the representation theory of the Monster group to modular functions; the supplied passage does not state whether this formulation has been proved.