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

$K$ is the global number field defined by \( x^{2} - 31 \)

$\times$ \(\chi_{ 124 } ( 1, ·)\) \(\chi_{ 124 } ( 123, ·)\)
\(\chi_{ 124 }(1, ·)\) \(\chi_{ 124 } ( 1, ·)\) \(\chi_{ 124 } ( 123, ·)\)
\(\chi_{ 124 }(123, ·)\) \(\chi_{ 124 } ( 123, ·)\) \(\chi_{ 124 } ( 1, ·)\)