The point-count conjecture for the double cover Q_3

Let $Q_3$ and $R_3$ be the quotient varieties associated with the involution $\iota_3$, and let $[Q_3]_p$ and $[R_3]_p$ denote their numbers of points over $\mathbb F_p$. Let $a_p$ be the eigenvalue of $T_p$ on the newform of weight $6$ and level $8$, and let $\phi$ denote the quadratic character appearing in the point-count formula. Point-count conjecture for $Q_3$ and $R_3$. For every odd prime $p$, one has

$$[R_3]_p=\sum_{i=0}^5p^i$$

and

$$[Q_3]_p=\sum_{i=0}^5p^i-a_p-\phi(-1)p^2.$$

The formulas are suggested by computations for small primes and by the predicted cohomological actions of the involution; their validity for all odd primes remains conjectural.