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

$K$ is the global number field defined by \(x^{3} \) \(\mathstrut -\mathstrut x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut +\mathstrut 8 \)

$\times$ \(\chi_{ 31 } ( 1, ·)\) \(\chi_{ 31 } ( 5, ·)\) \(\chi_{ 31 } ( 25, ·)\)
\(\chi_{ 31 }(1, ·)\) \(\chi_{ 31 } ( 1, ·)\) \(\chi_{ 31 } ( 5, ·)\) \(\chi_{ 31 } ( 25, ·)\)
\(\chi_{ 31 }(5, ·)\) \(\chi_{ 31 } ( 5, ·)\) \(\chi_{ 31 } ( 25, ·)\) \(\chi_{ 31 } ( 1, ·)\)
\(\chi_{ 31 }(25, ·)\) \(\chi_{ 31 } ( 25, ·)\) \(\chi_{ 31 } ( 1, ·)\) \(\chi_{ 31 } ( 5, ·)\)