from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(523, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,523))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(523\) | |
| Conductor: | \(523\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(29\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{29})$ |
| Fixed field: | Number field defined by a degree 29 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{523}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) |
| \(\chi_{523}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) |
| \(\chi_{523}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) |
| \(\chi_{523}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) |
| \(\chi_{523}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) |
| \(\chi_{523}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) |
| \(\chi_{523}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) |
| \(\chi_{523}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) |
| \(\chi_{523}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) |
| \(\chi_{523}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) |
| \(\chi_{523}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) |
| \(\chi_{523}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) |
| \(\chi_{523}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) |
| \(\chi_{523}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) |
| \(\chi_{523}(280,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) |
| \(\chi_{523}(285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) |
| \(\chi_{523}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) |
| \(\chi_{523}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) |
| \(\chi_{523}(368,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) |
| \(\chi_{523}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) |
| \(\chi_{523}(394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) |
| \(\chi_{523}(408,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) |
| \(\chi_{523}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) |
| \(\chi_{523}(465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) |
| \(\chi_{523}(473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) |
| \(\chi_{523}(490,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) |
| \(\chi_{523}(496,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) |
| \(\chi_{523}(520,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) |