The point-count conjecture for the double cover Q_2

Let $Q_2$ be the double cover considered in the paper, and let $[Q_2]_p$ denote its number of points over $\mathbb F_p$. Let $b_p$ be the Hecke eigenvalue of the newform of weight $4$ and level $8$. Point-count conjecture for $Q_2$. For every odd prime $p$, one has

$$[Q_2]_p=\sum_{i=0}^5p^i-pb_p.$$

The formula is established computationally in the supplied context for odd primes $p<20$ and is conjectured for all odd primes.