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

$K$ is the global number field defined by \( x^{2} - x - 772 \)

$\times$ \(\chi_{ 3089 } ( 1, ·)\) \(\chi_{ 3089 } ( 3088, ·)\)
\(\chi_{ 3089 }(1, ·)\) \(\chi_{ 3089 } ( 1, ·)\) \(\chi_{ 3089 } ( 3088, ·)\)
\(\chi_{ 3089 }(3088, ·)\) \(\chi_{ 3089 } ( 3088, ·)\) \(\chi_{ 3089 } ( 1, ·)\)