Point-count conjecture for the double octic O_6

Let $O_6$ be the double octic defined by the equation displayed in the source, and define

$$\alpha(p)=\begin{cases}1,&p\equiv1\pmod 8,\\0,&\text{otherwise.}\end{cases}$$

For each prime $p>3$, let $a_p$ be the eigenvalue of $T_p$ for the newform of weight $4$ and level $6$, and let $\left(\frac{-3}{p}\right)$ be the Legendre symbol. Point-count conjecture for $O_6$. The number of $F_p$-points of $O_6$ is

$$p^3+5p^2-(11+4\alpha(p)+\left(\frac{-3}{p}\right))p+1-a_p.$$

The statement is a computationally motivated modularity conjecture for a CalabiYau threefold arising from phi-four graph reduction; its resolution is not supplied in the source.