Modular congruence criterion for non-unirational surfaces

Let $Y$ be a surface satisfying the projective-resolution and congruence assumptions, and let $\widetilde{Y}$ be a fixed minimal smooth model. Suppose that for some $N\in\mathbb{N}$, a subgroup $\Gamma\leq\operatorname{PSL}_2(\mathbb{Z})$, and a positive integer $B$ with $\#\Sigma_0(B)\geq 10$, there is a newform $f=\sum_{k=1}^{\infty}b_kq^k\in S_3^{\mathrm{new}}(\Gamma(N))$ satisfying $b_p\equiv\#Y_p(\mathbb{F}_p)-1\pmod p$ for every $p\in\Sigma(B)$. Modular congruence criterion. If this condition holds, then $Y$ is birationally equivalent to a smooth surface $\widetilde{Y}$ with $h^{2,0}(\widetilde{Y})\geq 1$. In particular, $Y$ is not unirational. The criterion is intended as a point-counting consistency check for identifying surfaces whose smooth models have nontrivial holomorphic $2$-forms; the supplied text does not establish its status beyond this asserted implication.