from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,52,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,1620))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1620\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 81.g | 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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 27 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1620}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) |
| \(\chi_{1620}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) |
| \(\chi_{1620}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{27}\right)\) |
| \(\chi_{1620}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{1620}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{25}{27}\right)\) |
| \(\chi_{1620}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) |
| \(\chi_{1620}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{10}{27}\right)\) |
| \(\chi_{1620}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) |
| \(\chi_{1620}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{1620}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) |
| \(\chi_{1620}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) |
| \(\chi_{1620}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{1620}(1141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) |
| \(\chi_{1620}(1201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{1620}(1321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{1620}(1381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{1620}(1501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) |
| \(\chi_{1620}(1561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) |