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

$K$ is the global number field defined by \( x^{4} + 10 x^{2} + 20 \)

$\times$ \(\chi_{ 40 } ( 1, ·)\) \(\chi_{ 40 } ( 9, ·)\) \(\chi_{ 40 } ( 13, ·)\) \(\chi_{ 40 } ( 37, ·)\)
\(\chi_{ 40 }(1, ·)\) \(\chi_{ 40 } ( 1, ·)\) \(\chi_{ 40 } ( 9, ·)\) \(\chi_{ 40 } ( 13, ·)\) \(\chi_{ 40 } ( 37, ·)\)
\(\chi_{ 40 }(9, ·)\) \(\chi_{ 40 } ( 9, ·)\) \(\chi_{ 40 } ( 1, ·)\) \(\chi_{ 40 } ( 37, ·)\) \(\chi_{ 40 } ( 13, ·)\)
\(\chi_{ 40 }(13, ·)\) \(\chi_{ 40 } ( 13, ·)\) \(\chi_{ 40 } ( 37, ·)\) \(\chi_{ 40 } ( 9, ·)\) \(\chi_{ 40 } ( 1, ·)\)
\(\chi_{ 40 }(37, ·)\) \(\chi_{ 40 } ( 37, ·)\) \(\chi_{ 40 } ( 13, ·)\) \(\chi_{ 40 } ( 1, ·)\) \(\chi_{ 40 } ( 9, ·)\)