to be a subgroup of $SL(2,{{\mathbb Z}})$ (rather than a subgroup of $PSL(2,{{\mathbb Z}})$) that contains no element of order $2$. This is a reason we consider a lift $\tilde{\Gamma}$ of $\Gamma$ in our discussion.)

[**Remark 5 **]{}: [*It follows from Kodaira’s theory that when $G$ contains no elliptic elements and no elements of trace $-$2, all the singular fibers are above the cusps and are of type $I_n$, where $n$ is the width of the cusp. On the other hand, elements of trace $-2$ give rise to $I_n^*$-fibers above the cusps.*]{}

For the $\tilde{\Gamma}$’s of Proposition 4 the modular curve $X(\tilde{\Gamma})$ has genus $0$, and, since the sum of the widths of the cusps in Table 2 is always 24, each $E(\tilde{\Gamma})$ is an extremal K3 surface. The space $S_3(\tilde{\Gamma})$ of cusp forms of weight 3 for $\tilde{\Gamma}$ is therefore one-dimensional. Up to a square, the discriminant of the intersection form on the rank 2 motive of the transcendental cycles $T: =
T(E(\tilde{\Gamma})) =
          H^2(E(\tilde{\Gamma}),{{\mathbb Q}})/{\rm NS}
          (E(\tilde{\Gamma}))$ is given by

$$\label{delta}
\delta = \delta_k = -1,-3,-7,-2,-1,-2,-3,-1,-2$$

in cases $k = 1,\dots,9$ respectively. To see this one computes the discriminant of the (known) lattice $NS(E(\tilde{\Gamma}))$ and passes to the 