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In the 1960's, Birch proved that the traces of Frobenius for elliptic curvestaken at random over a large finite field is modeled by the semicirculardistribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogywith Birch's result, a recent paper by Ono, the author, and Saikia proved thatthe 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. Itis supported on $[-3,3]$ and has vertical asymptotes at $t=\pm1.$ Here we makethis result explicit. We prove that if $p\geq 5$ is prime and $-3\leq a<b\leq3,$ 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 afinite field $\mathbb{F}_p$ is large enough for the discrete histograms toreach any given height near $t=\pm1.$ To obtain these results, we make use ofthe theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.

Explicit Sato-Tate type distribution for a family of $K3$ surfaces