The point-count conjecture for the quotients of V_{32} by α_1 and α_2

Let $V_{32}$ be the double cover considered in the paper, and let $\alpha_1$ and $\alpha_2$ be the two involutions acting on it. Write $[V_{32}/\alpha_i]_p$ for the number of points of the quotient over $\mathbb F_p$, and let $a_{j,p}$ denote the relevant Hecke eigenvalues in weights $j=2,4,6$. Point-count conjecture for the $\alpha_i$-quotients. For every prime $p>2$, one has

$$[V_{32}/\alpha_1]_p=\sum_{i=0}^5p^i-a_{6,p}-p^2a_{2,p}$$

and

$$[V_{32}/\alpha_2]_p=\sum_{i=0}^5p^i-a_{6,p}-pa_{4,p}.$$

The formulas are motivated by point counts and the predicted decomposition of the cohomology; the supplied passage gives no proof for all primes.