  In the 1960's, Birch proved that the traces of Frobenius for elliptic curves
taken at random over a large finite field is modeled by the semicircular
distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy
with Birch's result, a recent paper by Ono, the author, and Saikia proved that
the limiting distribution of the normalized Frobenius traces $A_{\lambda}(p)$
of a certain family of $K3$ surfaces $X_\lambda$ with generic Picard rank $19$
is the $O(3)$ distribution. This distribution, which we denote by
$\frac{1}{4\pi}f(t),$ is quite different from the semicircular distribution. It
is supported on $[-3,3]$ and has vertical asymptotes at $t=\pm1.$ Here we make
this result explicit. We prove that if $p\geq 5$ is prime and $-3\leq a<b\leq
3,$ then $$ \left|\frac{\#\{\lambda\in\mathbb{F}_p
:A_{\lambda}(p)\in[a,b]\}}{p}-\frac{1}{4\pi}\int_a^b f(t)dt\right|\leq
\frac{110.84}{p^{1/4}}. $$ As a consequence, we are able to determine when a
finite field $\mathbb{F}_p$ is large enough for the discrete histograms to
reach any given height near $t=\pm1.$ To obtain these results, we make use of
the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.
