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

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 38 \)

$\times$ \(\chi_{ 151 } ( 1, ·)\) \(\chi_{ 151 } ( 150, ·)\)
\(\chi_{ 151 }(1, ·)\) \(\chi_{ 151 } ( 1, ·)\) \(\chi_{ 151 } ( 150, ·)\)
\(\chi_{ 151 }(150, ·)\) \(\chi_{ 151 } ( 150, ·)\) \(\chi_{ 151 } ( 1, ·)\)