examples
======================

Let $Y=E(\tilde{\Gamma})$ be one of the K3 surfaces of the previous Section, and let $g_Y =
g_{3,\Gamma}$ denote the corresponding cusp form of level $3$. If $\tilde{\Gamma}$ contains $\Gamma_1(M)$ (in Table 2 this happens in cases \#3, \#4, \#7, \#8, and \#9), and $M = M_Y$ is the maximal possible, then $M$ is the [*level*]{} of $g_Y$, and the [*newtype*]{} of $g_Y$ is the Dirichlet character $$\label{new}
\epsilon = \epsilon_Y,$$ of conductor $M$, so that $g_Y$ is in $S_3(\Gamma_0(M), \epsilon_Y)$. Notice that $\epsilon$ is odd (namely $\epsilon(-1) = -1$). Moreover, since the coefficients of $g_Y$ are integers, $\epsilon$ must be quadratic. (We will determine $\epsilon$ for our examples in Proposition 10 below.)

Let $E$ be an elliptic curve. We view the product $Y\times E$ as a family of abelian surfaces over $X(\tilde{\Gamma})$. The fiber $A_t
= A_{\Gamma,t}$ over each point $t\in
X(\tilde{\Gamma})$ is the product of the fiber $E_{\Gamma,t}$ of $E(\tilde{\Gamma})$ with $E$. Then we have the following

[**Theorem 6**]{}:

*(1) The product $Y\times E$ has the Hodge numbers $$h^{0,3}(Y\times E)=1,\, h^{1,0}(Y\times E) =
1\,\,\mbox{and}\,\, B_3(Y\times E)=44$$ (so that $Y\times E$ is not a Calabi–Yau threefold).*

\(2) The motive $T(Y\times E)=T(Y)\times H^1(E)$ is a submotive of $H^3(Y\times E)$. 