$\pi$ are now $1$.

For the third part we notice that the restriction of $\chi_i$ to $U_i$ in the decomposition (\[dec\]) above gives a Dirichlet character $\epsilon'_i$ on ${{\mathcal O}}_{K_i}$ of conductor $c_i{{\mathcal O}}_{K_i}$. The newtype Dirichlet character $\epsilon_i$ (on ${{\mathbb Z}}$) is then the product $\chi_{K_i}$ by the restriction of $\epsilon'_i$ to ${{\mathbb Z}}$. However, the conductors $c_i{{\mathcal O}}_{K_i}$ of $\chi_i$ are all $1$ or $2$, and the only character of ${{\mathbb Z}}$ of conductor $1$ or $2$ is trivial. Hence the newtype character of $h_i$ is $K_i$, concluding the proof of Proposition 10.

[**Defining equations for extremal K3 surfaces**]{}: We now discuss how to determine defining equations for the extremal K3 surfaces in Theorem 7. This problem has been getting a considerable attention lately, for instance, Shioda \[Sh3\] and (independently and by a different method by Y. Iron \[I\]) have determined a defining equation for the semi-stable elliptic K3 surface with singular fibers of type $I_1,I_1,I_1,I_1,I_1,I_{19}$ whose existence was established in Miranda and Persson \[MP\] (this is given as $\#1$ in their list). As we shall see, our examples can be determined by a more classical method.

There are several cases where defining equations can be found in 