from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,1058))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1058\) | |
| Conductor: | \(529\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 529.e | 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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1058}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{1058}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{1058}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{1058}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{1058}(231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{1058}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{1058}(323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{1058}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{1058}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{1058}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{1058}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{1058}(553,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{1058}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{1058}(645,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) |
| \(\chi_{1058}(691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{1058}(737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{1058}(783,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{1058}(829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{1058}(875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{1058}(921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{1058}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{1058}(1013,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) |