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

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut -\mathstrut 10 \)

$\times$ \(\chi_{ 41 } ( 1, ·)\) \(\chi_{ 41 } ( 40, ·)\)
\(\chi_{ 41 }(1, ·)\) \(\chi_{ 41 } ( 1, ·)\) \(\chi_{ 41 } ( 40, ·)\)
\(\chi_{ 41 }(40, ·)\) \(\chi_{ 41 } ( 40, ·)\) \(\chi_{ 41 } ( 1, ·)\)