K3 surfaces with Picard number 19, see e. g. Besser and Livné \[BL\].) Thus one can use a result of Tate to lift each $\ell$-adic representation to the associated spin cover, which is the multiplicative group of some quaternion algebra over ${{\mathbb Q}}$. If the spin representation is modular (which should always be the case), then it is associated to a cusp form $h$ of weight $2$ on $GL(2)$, so that Symm$^2 h$ realizes $T(Y)$. Let $g_E$ be again the weight $2$ cusp form associated with $E$. It follows, by work of Gelbart and Jacquet, that $T(Y)$ is realized by an automorphic representation on $GL(3,{{\mathbb Q}})$. Hence, by the work of Kim and Shahidi (\[KS\]), $T(Y)\times E$ is realized by an automorphic form on $GL(6,{{\mathbb Q}})$. In particular, $L(\text{Symm}^2h \otimes g_E,s)$ has the expected analytic properties.*]{}

To construct our promised examples, let $X=
X_\Gamma{\rightarrow}X(\tilde{\Gamma})$ be the associated Kummer family, in which we divide each fiber $A_t$ of $Y(\tilde{\Gamma})\times E$ by $\pm 1$ and then blow up the locus of points of order $2$. We now have the following

[**Theorem 7 **]{}: [*In the Examples \#1, \#2, \#5, and \#6 of Table 2 the resulting $X$ is a smooth Calabi–Yau threefold. 