Hilbert modular surfaces are discussed in various papers by Hirzebruch. Following [HZ] (and their notations), one obtains Hilbert modular surfaces by the action of Hilbert modular group on $\mathcal{H}\times \mathcal{H}$ and $\mathcal{H}\times \mathcal{H}^{-}$, where $\mathcal{H}$ is the upper half plane and $\mathcal{H}^{-}$ is the lower half plane. In [HZ], they denote them as $Y(D)$ and $Y_{-}({D})$ respectively. In [EK], the authors studied and computed rational models of $Y_{-}(D)$ for $1<D<100$, and in particular, they computed the Picard number when $Y_{-}(D)$ is a K3 surface. On the other hand, their method seems to not cover the case $Y(D)$. And here is my question: can we compute their Picard number when $Y(D)$ is a K3 surface (i.e., when $D=29, 37, 40, 41, 44, 56, 57, 69, 105$) without necessarily construct rational models?
[EK] Elkies, Noam; Kumar, Abhinav, K3 surfaces and equations for Hilbert modular surfaces, Algebra Number Theory 8 (2014), no. 10, 2297–2411.
[HZ] Hirzebruch, F.; Zagier, D., Classification of Hilbert modular surfaces, Iwanami Shoten Publishers, Tokyo, 1977, pp. 43–77.
