from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(841, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(30,841))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(841\) | |
| Conductor: | \(841\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{841}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) |
| \(\chi_{841}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) |
| \(\chi_{841}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) |
| \(\chi_{841}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) |
| \(\chi_{841}(146,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) |
| \(\chi_{841}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) |
| \(\chi_{841}(204,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) |
| \(\chi_{841}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) |
| \(\chi_{841}(262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) |
| \(\chi_{841}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) |
| \(\chi_{841}(320,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) |
| \(\chi_{841}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) |
| \(\chi_{841}(378,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) |
| \(\chi_{841}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) |
| \(\chi_{841}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) |
| \(\chi_{841}(465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) |
| \(\chi_{841}(494,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) |
| \(\chi_{841}(523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) |
| \(\chi_{841}(552,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) |
| \(\chi_{841}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) |
| \(\chi_{841}(610,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) |
| \(\chi_{841}(639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) |
| \(\chi_{841}(668,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) |
| \(\chi_{841}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) |
| \(\chi_{841}(726,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) |
| \(\chi_{841}(755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) |
| \(\chi_{841}(784,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) |
| \(\chi_{841}(813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) |