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

$K$ is the global number field defined by \( x^{2} - 1126 \)

$\times$ \(\chi_{ 4504 } ( 1, ·)\) \(\chi_{ 4504 } ( 2251, ·)\)
\(\chi_{ 4504 }(1, ·)\) \(\chi_{ 4504 } ( 1, ·)\) \(\chi_{ 4504 } ( 2251, ·)\)
\(\chi_{ 4504 }(2251, ·)\) \(\chi_{ 4504 } ( 2251, ·)\) \(\chi_{ 4504 } ( 1, ·)\)