  Let $M$ be a 3-manifold with a finite set $X(M)$ of conjugacy classes of
representations $\rho:\pi_1(M)\to$SU$_2$. We study here the distribution of the
values of the Chern-Simons function CS$:X(M)\to \mathbb{R}/2\pi\mathbb{Z}$. We
observe in some examples that it resembles the distribution of quadratic
residues. In particular for specific sequences of $3$-manifolds, the invariants
tends to become equidistributed on the circle with white noise fluctuations of
order $|X(M)|^{-1/2}$. We prove that for a manifold with toric boundary the
Chern-Simons invariants of the Dehn fillings $M_{p/q}$ have the same behaviour
when $p$ and $q$ go to infinity and compute fluctuations at first order.
