from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(283920, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,39,78,78,26,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(59,283920))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(283920\) | |
| Conductor: | \(283920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(156\) | 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: | $\Q(\zeta_{156})$ |
| Fixed field: | Number field defined by a degree 156 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{283920}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) |
| \(\chi_{283920}(13739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{53}{156}\right)\) |
| \(\chi_{283920}(17699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{115}{156}\right)\) |
| \(\chi_{283920}(19619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{59}{156}\right)\) |
| \(\chi_{283920}(21899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{25}{156}\right)\) |
| \(\chi_{283920}(39539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{7}{156}\right)\) |
| \(\chi_{283920}(41459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{47}{156}\right)\) |
| \(\chi_{283920}(43739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{133}{156}\right)\) |
| \(\chi_{283920}(57419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{77}{156}\right)\) |
| \(\chi_{283920}(61379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{55}{156}\right)\) |
| \(\chi_{283920}(63299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) |
| \(\chi_{283920}(65579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{85}{156}\right)\) |
| \(\chi_{283920}(79259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{89}{156}\right)\) |
| \(\chi_{283920}(83219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) |
| \(\chi_{283920}(85139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{23}{156}\right)\) |
| \(\chi_{283920}(87419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{37}{156}\right)\) |
| \(\chi_{283920}(101099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{101}{156}\right)\) |
| \(\chi_{283920}(105059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{151}{156}\right)\) |
| \(\chi_{283920}(106979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{11}{156}\right)\) |
| \(\chi_{283920}(109259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{145}{156}\right)\) |
| \(\chi_{283920}(122939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{113}{156}\right)\) |
| \(\chi_{283920}(126899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{43}{156}\right)\) |
| \(\chi_{283920}(128819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{155}{156}\right)\) |
| \(\chi_{283920}(131099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{97}{156}\right)\) |
| \(\chi_{283920}(144779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{125}{156}\right)\) |
| \(\chi_{283920}(152939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{49}{156}\right)\) |
| \(\chi_{283920}(166619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{137}{156}\right)\) |
| \(\chi_{283920}(170579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{139}{156}\right)\) |
| \(\chi_{283920}(172499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{131}{156}\right)\) |
| \(\chi_{283920}(174779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{1}{156}\right)\) |
| \(\chi_{283920}(188459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{149}{156}\right)\) |