Abstract In the 1960s, 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 SatoâTate for non-CM elliptic curves). In analogy with Birchâs result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces A Îť ⢠( p ) {A_{\lambda}(p)} of a certain family of K ⢠3 {K3} surfaces X Îť {X_{\lambda}} with generic Picard rank 19 is the O ⢠( 3 ) {O(3)} distribution. This distribution, which we denote by 1 4 â˘ Ď â˘ f ⢠( t ) {\frac{1}{4\pi}f(t)} , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] {[-3,3]} and has vertical asymptotes at t = Âą 1 {t=\pm 1} . Here we make this result explicit. We prove that if p ⼠5 {p\geq 5} is prime and - 3 ⤠a < b ⤠3 {-3\leq a<b\leq 3} , then | # ⢠{ Îť â đ˝ p : A Îť ⢠( p ) â [ a , b ] } p - 1 4 â˘ Ď â˘ âŤ a b f ⢠( t ) ⢠đ t | ⤠98.28 p 1 / 4 . \biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-% \frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field đ˝ p {\mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near t = Âą 1 {t=\pm 1} . To obtain these results, we make use of the theory of RankinâCohen brackets in the theory of harmonic Maass forms.