from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(311, base_ring=CyclotomicField(62))
M = H._module
chi = DirichletCharacter(H, M([28]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,311))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(311\) | |
| Conductor: | \(311\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(31\) | 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: | $\Q(\zeta_{31})$ |
| Fixed field: | Number field defined by a degree 31 polynomial |
First 30 of 30 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{311}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(1\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) |
| \(\chi_{311}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(1\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) |
| \(\chi_{311}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(1\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) |
| \(\chi_{311}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(1\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) |
| \(\chi_{311}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(1\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) |
| \(\chi_{311}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(1\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) |
| \(\chi_{311}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(1\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) |
| \(\chi_{311}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(1\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) |
| \(\chi_{311}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(1\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) |
| \(\chi_{311}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(1\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) |
| \(\chi_{311}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(1\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) |
| \(\chi_{311}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(1\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) |
| \(\chi_{311}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(1\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) |
| \(\chi_{311}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(1\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) |
| \(\chi_{311}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(1\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) |
| \(\chi_{311}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(1\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) |
| \(\chi_{311}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(1\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) |
| \(\chi_{311}(146,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(1\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) |
| \(\chi_{311}(168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(1\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) |
| \(\chi_{311}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(1\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) |
| \(\chi_{311}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(1\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) |
| \(\chi_{311}(224,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(1\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) |
| \(\chi_{311}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(1\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) |
| \(\chi_{311}(234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(1\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{8}{31}\right)\) |
| \(\chi_{311}(243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(1\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) |
| \(\chi_{311}(250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(1\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{20}{31}\right)\) |
| \(\chi_{311}(260,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(1\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) |
| \(\chi_{311}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(1\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{10}{31}\right)\) |
| \(\chi_{311}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(1\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) |
| \(\chi_{311}(300,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(1\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{9}{31}\right)\) |