  For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the
adele ring $\mathbb{A}$, we study the distribution of angles
$\theta_\pi(\mathfrak{p})$ coming from Satake parameters corresponding to
unramified $\pi_\mathfrak{p}$ where $\pi_\mathfrak{p}$ comes from a global
$\pi$ ranging over a certain finite set $\Pi_{\underline{k}}(\mathfrak{n})$ of
cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central
character. For such a representation $\pi$, it is known that the angles
$\theta_\pi(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an
interval $I\subseteq [0,\pi]$, we prove a central limit theorem for the number
of angles $\theta_\pi(\mathfrak{p})$ that lie in $I$, as
$\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a
squarefree integral ideal, and that the components in the weight vector
$\underline{k}$ grow suitably fast as a function of $x$.
