structural:expansion, information:classification
Real characters

χ (a) {\displaystyle \chi (a)} is real or quadratic if all of its values are real (they must be 0, ± 1 {\displaystyle 0,\;\pm 1}); otherwise it is complex or imaginary.

χ {\displaystyle \chi } is real if and only if χ 2 = χ 0 {\displaystyle \chi ^{2}=\chi _{0}}; χ m, k {\displaystyle \chi _{m,k}} is real if and only if k 2 ≡ 1 (mod m) {\displaystyle k^{2}\equiv 1{\pmod {m}}}; in particular, χ m, − 1 {\displaystyle \chi _{m,-1}} is real and non-principal.

Dirichlet's original proof that L (1, χ) ≠ 0 {\displaystyle L(1,\chi)\neq 0} (which was only valid for prime moduli) took two different forms depending on whether χ {\displaystyle \chi } was real or not. His later proof, valid for all moduli, was based on his class number formula.

Real characters are Kronecker symbols; for example, the principal character can be written χ m, 1 = (m 2 ∙) {\displaystyle \chi _{m,1}=\left({\frac {m^{2}}{\bullet }}\right)}.

The real characters in the examples are: