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

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut -\mathstrut 251 \)

$\times$ \(\chi_{ 1004 } ( 1, ·)\) \(\chi_{ 1004 } ( 1003, ·)\)
\(\chi_{ 1004 }(1, ·)\) \(\chi_{ 1004 } ( 1, ·)\) \(\chi_{ 1004 } ( 1003, ·)\)
\(\chi_{ 1004 }(1003, ·)\) \(\chi_{ 1004 } ( 1003, ·)\) \(\chi_{ 1004 } ( 1, ·)\)