$\Gamma_1(7)$. This case differs from the examples considered in our paper with the main difference being the fact that the $2$-torsion points do not decompose into four sections, leading to non-semi-stable fibrations. But it still gives rise to a Calabi–Yau threefold (Theorem 2.2 of Hulek and Verrill \[HV\]), and the modularity question can still be considered, and this is exactly what Hulek and Verrill deals with in the supplement \[HV\] to this article.

Extremal congruence K3 surfaces
===============================

The torsion-free genus zero congruence subgroups of $PSL(2,{{\mathbb Z}})$ of index $24$ were classified in Sebbar \[Se\]. There are precisely nine conjugacy classes of such congruence subgroups.

The second column in the following Table 2 gives the complete list of the torsion-free congruence subgroups of $PSL(2,{{\mathbb Z}})$ of index $24$ up to conjugacy. Each has precisely 6 cusps. The third column in the table gives the widths of these cusps.

$$\begin{array}{|c|c|c|} \hline\hline
\#& \mbox{The group $\Gamma$} &
\mbox{Widths of the cusps}  \\
\hline
1& \Gamma(4) & 4,4,4,4,4,4  \\ \hline
2& \Gamma_0(3)\cap \Gamma(2) & 6,6,6,2,2,2  \\
\hline
3& \Gamma_1(7) & 7,7,7,1,1,1  \\ \hline
4& \Gamma_1(8) & 8,8,4,2,1,1  \\ \hline
5& \Gamma_0(8)\cap \Gamma(2) & 8,8,2,2,2,2  \\
\hline
6& \Gamma_1(8;4,1,2) & 8,4,4,4,2,2  \\ \hline
7& 