If $E$ and $Y$ are defined over ${{\mathbb Q}}$, this submotive is modular, in the sense that its $L$-series is associated to a cusp form $g_Y$ on $GL(2,{{\mathbb Q}}(\sqrt{\delta}\,))$. Let $g_E$ be the cusp form of weight $2$ associated to $E$ by Wiles et. al. (\[W\]). Let $A(p)$ (respectively $B(p)$) be the $p$th Fourier coefficient of $g_E$ (respectively of $g_Y$), and let $\epsilon_Y$ be the newtype character of $g_Y$ (see Section 3). Then for any good prime $p$, the local Euler factor $L_p(s)$ of the $L$-series $L(T(Y\times E),s) = L(g_E\otimes g_Y,s)$ is $$1 - A(p)B(p)p^{-s} + (B(p)^2+\epsilon_Y(p)pA(p)^2-2p^2
 \epsilon_Y(p))p^{1-2s}
  - A(p)B(p)\epsilon_Y(p)p^{3-3s} + p^{6-4s}.$$

[**Proof**]{}: The statements about the Hodge and Betti numbers follow from the Künneth formula. Since $T(Y)$ is a factor of $H^2(Y)$, it follows again from the Künneth formula that $T(Y)\times H^1(E)$ is a factor of $H^3(Y\times E)$.

For the second part, we know that $g_Y$ is a CM form. Hence it is induced from an algebraic Hecke character $\chi = \chi_Y$ of the imaginary quadratic field $F = K_i$. Let $\chi_G$ be the compatible system of $1$-dimensional $\ell$-adic representations of $G_F =
\text{Gal}(\overline{{\mathbb Q}}/F)$ corresponding to $\chi$. Then the $2$-dimensional Galois representation associated to $T(Y)$ is ind$_{G_F}^{G_{{\mathbb Q}}}\chi_G$. 