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

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut +\mathstrut 89 \)

$\times$ \(\chi_{ 356 } ( 1, ·)\) \(\chi_{ 356 } ( 355, ·)\)
\(\chi_{ 356 }(1, ·)\) \(\chi_{ 356 } ( 1, ·)\) \(\chi_{ 356 } ( 355, ·)\)
\(\chi_{ 356 }(355, ·)\) \(\chi_{ 356 } ( 355, ·)\) \(\chi_{ 356 } ( 1, ·)\)