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

$K$ is the global number field defined by \( x^{2} - x + 171 \)

$\times$ \(\chi_{ 683 } ( 1, ·)\) \(\chi_{ 683 } ( 682, ·)\)
\(\chi_{ 683 }(1, ·)\) \(\chi_{ 683 } ( 1, ·)\) \(\chi_{ 683 } ( 682, ·)\)
\(\chi_{ 683 }(682, ·)\) \(\chi_{ 683 } ( 682, ·)\) \(\chi_{ 683 } ( 1, ·)\)