from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(764, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,18]))
chi.galois_orbit()
[g,chi] = znchar(Mod(107,764))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(764\) | |
| Conductor: | \(764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{764}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(-1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{764}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(-1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{764}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(-1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{764}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(-1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{764}(327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(-1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) |
| \(\chi_{764}(351,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(-1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{764}(371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(-1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{764}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(-1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{764}(407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(-1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{764}(451,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(-1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{764}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(-1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{764}(507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(-1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{764}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(-1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) |
| \(\chi_{764}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(-1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{764}(579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(-1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{764}(603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(-1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{764}(723,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(-1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) |
| \(\chi_{764}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(-1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) |