how to construct examples corresponding to Example \#3 in Table 2, for which $\tilde{\Gamma} = \Gamma_1(7)$. We also do not know whether our examples are the only ones.*

\(2) It is an interesting exercise to compute the full $L$-series of our examples. The results are as follows: Let $N_+$ (respectively $N_-$) be the motive of algebraic cycles on $Y$ invariant (respectively anti-invariant) by $\pm
1$ acting on the elliptic fibrations of $Y$. Let $n_{\pm}$ be the rank of $N_\pm$. Then $n_+ + n_-
= 20$, and if we let $\chi_{\delta^{\prime}}$ denote the quadratic character cut by ${{\mathbb Q}}(\sqrt{\delta^{\prime}}\,)$ (not necessarily the same quadratic field pre-determined by the modular group corresponding to the surface), then $N_+= {{\mathbb Z}}(1)^{n^{\prime}_+}\oplus{{\mathbb Z}}(\chi_{\delta^{\prime}}(1))^{n^{\prime\prime}_+}$ and $N_- = {{\mathbb Z}}(1)^{n^{\prime}_-}\oplus {{\mathbb Z}}(\chi_\delta^{\prime}(1))^{n^{\prime\prime}_-}$. Here $n^{\prime}_{+}$ (resp. $n^{\prime\prime}_+$) denotes the number of cycles defined over ${{\mathbb Q}}$ (resp. ${{\mathbb Q}}(\delta^{\prime})$) and similarly for $n^{\prime}_-$ (resp. $n^{\prime\prime}_-$). We have $n_{\pm}=
(n^{\prime}+n^{\prime\prime})_{\pm}$. Then we have

$$\begin{aligned}
L(H^0,s) & = & L({{\mathbb Z}},s) = \zeta(s) \\
L(H^1,s) & = & 1 \\
L(H^2,s) &= & L(H^2({{{\mathbb P}}}^1\times{{{\mathbb P}}}^1),s)L({{\mathbb Z}}(1),s)L(N_+,s)\\
          &=& \zeta(s-1)^{16}\zeta(s-1)^{1+n_+^{\prime}}
L({{\mathbb Q}}\otimes\chi_{\delta^{\prime}},s-1)^{n_+^{\prime\prime}}\\
L(H^3,s) & = & L(T(Y)\otimes H^1(E),s)L(N_-\times H^1(E),s) \\
\quad    & = & L(g_3\otimes g_2,s)L(E, s-1)^{n_-^{\prime}}
\prod_{\delta^{\prime}}L(E\otimes\chi_{{\delta}^{\prime}},s-1)^{n_-^{\prime\prime}}\end{aligned}$$

(The higher cohomologies 