Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut -\mathstrut 3 \)

$\times$ \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 12, ·)\)
\(\chi_{ 13 }(1, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 12, ·)\)
\(\chi_{ 13 }(12, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 1, ·)\)