orthogonal complement in $H^2(E(\tilde{\Gamma}),{{\mathbb Q}})$. For details, see e. g. Besser-Livné \[BS\]. By \[L2\] it follows that the normalized newform $g_{3,\Gamma}$ generating $S_3(\tilde{\Gamma})$ has CM by ${{\mathbb Q}}(\sqrt{\delta}\,)$.

To each of our 9 examples there is a naturally associated moduli problem of classifying (generalized) elliptic curves with a given level structure (Katz and Mazur \[KM\]). Each of these moduli problems refines the respective moduli problem $Y_1(M)$, of classifying elliptic curves with a point of order $M$, where $M$ is as above. Since $M\geq 4$, these moduli problems are all represented by universal families $E(\tilde{\Gamma})/X(\tilde{\Gamma})/{{\mathbb Z}}[1/M]$ (see Katz–Mazur \[KM\]). The geometric fibers are geometrically connected in all these examples, and their compactified fibers over ${{\mathbb C}}$ are the corresponding elliptic modular surfaces above.

We shall now compute the $L$-series $L(T,s)$ of the transcendental cycles. By the Eichler–Shimura Isomorphism, this is the parabolic cohomology $$\tilde{H}: =
\tilde{H}{\vphantom{H}}^1(
X(\tilde{\Gamma})\times_{{{\mathbb Z}}[1/M]}\overline{{{\mathbb Q}}},
R^1(E(\tilde{\Gamma})\rightarrow
X(\tilde{\Gamma}))).$$ Moreover, Deligne proved (\[D\]) the Eichler–Shimura congruence relation $$\text{Frob}_p + \text{Frob}'_p = T_p
\qquad\text{for any $p\not|M$},$$ where $T_p$ is the $p$-th Hecke operator on $S_3(\tilde{\Gamma})$. This is the same as the $p$-th Fourier coefficient of the normalized newform $g_{3,\Gamma}$. Summarizing, we proved $$L(T(X(\tilde{\Gamma}),s) = L(g_{3,\Gamma},s).$$

Explicit Weierstrass equations for the 