from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(216))
M = H._module
chi = DirichletCharacter(H, M([108,189,44]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,5184))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5184\) | |
| Conductor: | \(2592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(216\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2592.ci | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{216})$ |
| Fixed field: | Number field defined by a degree 216 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5184}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{216}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{113}{216}\right)\) | \(e\left(\frac{163}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{35}{216}\right)\) | \(e\left(\frac{31}{54}\right)\) |
| \(\chi_{5184}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{216}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{37}{216}\right)\) | \(e\left(\frac{191}{216}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{151}{216}\right)\) | \(e\left(\frac{35}{54}\right)\) |
| \(\chi_{5184}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{216}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{61}{216}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{29}{216}\right)\) | \(e\left(\frac{1}{54}\right)\) |
| \(\chi_{5184}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{216}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{139}{216}\right)\) | \(e\left(\frac{17}{216}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{1}{216}\right)\) | \(e\left(\frac{41}{54}\right)\) |
| \(\chi_{5184}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{216}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{173}{216}\right)\) | \(e\left(\frac{175}{216}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{23}{216}\right)\) | \(e\left(\frac{25}{54}\right)\) |
| \(\chi_{5184}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{216}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{25}{216}\right)\) | \(e\left(\frac{59}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{67}{216}\right)\) | \(e\left(\frac{47}{54}\right)\) |
| \(\chi_{5184}(455,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{216}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{203}{216}\right)\) | \(e\left(\frac{73}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{17}{216}\right)\) | \(e\left(\frac{49}{54}\right)\) |
| \(\chi_{5184}(551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{216}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{127}{216}\right)\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{133}{216}\right)\) | \(e\left(\frac{53}{54}\right)\) |
| \(\chi_{5184}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{216}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{17}{216}\right)\) | \(e\left(\frac{187}{216}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{19}{54}\right)\) |
| \(\chi_{5184}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{216}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{13}{216}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{199}{216}\right)\) | \(e\left(\frac{5}{54}\right)\) |
| \(\chi_{5184}(743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{216}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{47}{216}\right)\) | \(e\left(\frac{85}{216}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{5}{216}\right)\) | \(e\left(\frac{43}{54}\right)\) |
| \(\chi_{5184}(839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{216}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{115}{216}\right)\) | \(e\left(\frac{185}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{49}{216}\right)\) | \(e\left(\frac{11}{54}\right)\) |
| \(\chi_{5184}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{216}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{77}{216}\right)\) | \(e\left(\frac{199}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{215}{216}\right)\) | \(e\left(\frac{13}{54}\right)\) |
| \(\chi_{5184}(983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{216}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{1}{216}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{115}{216}\right)\) | \(e\left(\frac{17}{54}\right)\) |
| \(\chi_{5184}(1031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{216}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{107}{216}\right)\) | \(e\left(\frac{97}{216}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{209}{216}\right)\) | \(e\left(\frac{37}{54}\right)\) |
| \(\chi_{5184}(1127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{216}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{103}{216}\right)\) | \(e\left(\frac{53}{216}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{181}{216}\right)\) | \(e\left(\frac{23}{54}\right)\) |
| \(\chi_{5184}(1175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{216}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{211}{216}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{203}{216}\right)\) | \(e\left(\frac{7}{54}\right)\) |
| \(\chi_{5184}(1271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{205}{216}\right)\) | \(e\left(\frac{95}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{31}{216}\right)\) | \(e\left(\frac{29}{54}\right)\) |
| \(\chi_{5184}(1319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{216}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{167}{216}\right)\) | \(e\left(\frac{109}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{197}{216}\right)\) | \(e\left(\frac{31}{54}\right)\) |
| \(\chi_{5184}(1415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{216}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{91}{216}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{97}{216}\right)\) | \(e\left(\frac{35}{54}\right)\) |
| \(\chi_{5184}(1463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{216}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{197}{216}\right)\) | \(e\left(\frac{7}{216}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{191}{216}\right)\) | \(e\left(\frac{1}{54}\right)\) |
| \(\chi_{5184}(1559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{216}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{193}{216}\right)\) | \(e\left(\frac{179}{216}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{163}{216}\right)\) | \(e\left(\frac{41}{54}\right)\) |
| \(\chi_{5184}(1607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{216}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{121}{216}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{185}{216}\right)\) | \(e\left(\frac{25}{54}\right)\) |
| \(\chi_{5184}(1703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{216}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{79}{216}\right)\) | \(e\left(\frac{5}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{13}{216}\right)\) | \(e\left(\frac{47}{54}\right)\) |
| \(\chi_{5184}(1751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{216}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{41}{216}\right)\) | \(e\left(\frac{19}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{179}{216}\right)\) | \(e\left(\frac{49}{54}\right)\) |
| \(\chi_{5184}(1847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{216}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{181}{216}\right)\) | \(e\left(\frac{47}{216}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{79}{216}\right)\) | \(e\left(\frac{53}{54}\right)\) |
| \(\chi_{5184}(1895,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{216}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{71}{216}\right)\) | \(e\left(\frac{133}{216}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{173}{216}\right)\) | \(e\left(\frac{19}{54}\right)\) |
| \(\chi_{5184}(1991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{216}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{67}{216}\right)\) | \(e\left(\frac{89}{216}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{145}{216}\right)\) | \(e\left(\frac{5}{54}\right)\) |
| \(\chi_{5184}(2039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{216}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{31}{216}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{167}{216}\right)\) | \(e\left(\frac{43}{54}\right)\) |
| \(\chi_{5184}(2135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{169}{216}\right)\) | \(e\left(\frac{131}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{211}{216}\right)\) | \(e\left(\frac{11}{54}\right)\) |
| \(\chi_{5184}(2183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{216}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{131}{216}\right)\) | \(e\left(\frac{145}{216}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{161}{216}\right)\) | \(e\left(\frac{13}{54}\right)\) |