It might be helpful to recall the six rigid Calabi–Yau threefolds in Theorem 2. These six rigid Calabi–Yau threefolds are obtained (by Schoen \[S\]; see also Beauville \[B\]) as the self-fiber products of stable families of elliptic curves admitting exactly four singular fibers. The base curve is a rational modular curve and correspond to the torsion-free genus zero congruence subgroups $\Gamma$ of $PSL(2,{{\mathbb Z}})$ in Table 1. Note that the $4$-tuples of natural numbers appearing in the second column add up to $12$, which is the index of the modular group $\Gamma$ in $PSL(2,{{\mathbb Z}})$.

In \[STZ\] the authors indicate one example for the non-rigid extremal case. It is related to $\Gamma(4)$, which is a torsion-free congruence subgroup of genus $0$ and index $24$ in $PSL(2,{{\mathbb Z}})$. The list of torsion-free congruence subgroups of genus $0$ and index 24 in $PSL(2,{{\mathbb Z}})$ is known (see Sebbar \[Se\], and Table 2 below). In this paper we will show that most of them give rise to a similar collection of examples. In each of these cases we will compute the interesting part of the $L$-series of the third cohomology of an appropriate natural model over ${{\mathbb Q}}$ in terms of automorphic forms.

