from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1113, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,26,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,1113))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1113\) | |
| Conductor: | \(371\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 371.q | 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: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 39 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1113}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) |
| \(\chi_{1113}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{1113}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{1113}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{1113}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{1113}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) |
| \(\chi_{1113}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) |
| \(\chi_{1113}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) |
| \(\chi_{1113}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) |
| \(\chi_{1113}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) |
| \(\chi_{1113}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) |
| \(\chi_{1113}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) |
| \(\chi_{1113}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{1113}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) |
| \(\chi_{1113}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{1113}(646,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{1113}(823,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) |
| \(\chi_{1113}(844,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) |
| \(\chi_{1113}(970,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{1113}(982,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) |
| \(\chi_{1113}(1003,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) |
| \(\chi_{1113}(1054,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{1113}(1075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{1113}(1096,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) |