  We study the large $N$ expansion of twisted partition functions of 3d
$\mathcal{N}=2$ superconformal field theories arising from $N$ M5-branes
wrapped on a hyperbolic 3-manifold, $M_3$. Via the 3d-3d correspondence, the
partition functions of these 3d ${\cal N}=2$ superconformal field theories are
related to simple topological invariants on the 3-manifold. The partition
functions can be expressed using only classical and one-loop perturbative
invariants of $PSL(N,\mathbb{C})$ Chern-Simons theory around irreducible flat
connections on $M_3$. Using mathematical results on the asymptotics of the
invariants, we compute the twisted partition functions in the large $N$ limit
including perturbative corrections to all orders in $1/N$. Surprisingly, the
perturbative expansion terminates at finite order. The leading part of the
partition function is of order $N^3$ and agrees with the Bekenstein-Hawking
entropy of the dual black holes. The subleading part, in particular the $\log
N$-terms in the field theory partition function is found to precisely match the
one-loop quantum corrections in the dual eleven dimensional supergravity. The
field theory results of other terms in $1/N$ provide a stringent prediction for
higher order corrections in the holographic dual, which is M-theory.
