from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(486))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,729))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(729\) | |
| Conductor: | \(729\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(243\) | 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: | $\Q(\zeta_{243})$ |
| Fixed field: | Number field defined by a degree 243 polynomial (not computed) |
First 31 of 162 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{729}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{243}\right)\) | \(e\left(\frac{2}{243}\right)\) | \(e\left(\frac{23}{243}\right)\) | \(e\left(\frac{151}{243}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{40}{243}\right)\) | \(e\left(\frac{89}{243}\right)\) | \(e\left(\frac{152}{243}\right)\) | \(e\left(\frac{4}{243}\right)\) |
| \(\chi_{729}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{243}\right)\) | \(e\left(\frac{151}{243}\right)\) | \(e\left(\frac{157}{243}\right)\) | \(e\left(\frac{101}{243}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{104}{243}\right)\) | \(e\left(\frac{37}{243}\right)\) | \(e\left(\frac{55}{243}\right)\) | \(e\left(\frac{59}{243}\right)\) |
| \(\chi_{729}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{166}{243}\right)\) | \(e\left(\frac{89}{243}\right)\) | \(e\left(\frac{173}{243}\right)\) | \(e\left(\frac{37}{243}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{79}{243}\right)\) | \(e\left(\frac{194}{243}\right)\) | \(e\left(\frac{203}{243}\right)\) | \(e\left(\frac{178}{243}\right)\) |
| \(\chi_{729}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{243}\right)\) | \(e\left(\frac{4}{243}\right)\) | \(e\left(\frac{46}{243}\right)\) | \(e\left(\frac{59}{243}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{80}{243}\right)\) | \(e\left(\frac{178}{243}\right)\) | \(e\left(\frac{61}{243}\right)\) | \(e\left(\frac{8}{243}\right)\) |
| \(\chi_{729}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{142}{243}\right)\) | \(e\left(\frac{41}{243}\right)\) | \(e\left(\frac{107}{243}\right)\) | \(e\left(\frac{58}{243}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{91}{243}\right)\) | \(e\left(\frac{2}{243}\right)\) | \(e\left(\frac{200}{243}\right)\) | \(e\left(\frac{82}{243}\right)\) |
| \(\chi_{729}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{243}\right)\) | \(e\left(\frac{46}{243}\right)\) | \(e\left(\frac{43}{243}\right)\) | \(e\left(\frac{71}{243}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{191}{243}\right)\) | \(e\left(\frac{103}{243}\right)\) | \(e\left(\frac{94}{243}\right)\) | \(e\left(\frac{92}{243}\right)\) |
| \(\chi_{729}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{172}{243}\right)\) | \(e\left(\frac{101}{243}\right)\) | \(e\left(\frac{68}{243}\right)\) | \(e\left(\frac{214}{243}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{76}{243}\right)\) | \(e\left(\frac{242}{243}\right)\) | \(e\left(\frac{143}{243}\right)\) | \(e\left(\frac{202}{243}\right)\) |
| \(\chi_{729}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{243}\right)\) | \(e\left(\frac{34}{243}\right)\) | \(e\left(\frac{148}{243}\right)\) | \(e\left(\frac{137}{243}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{194}{243}\right)\) | \(e\left(\frac{55}{243}\right)\) | \(e\left(\frac{154}{243}\right)\) | \(e\left(\frac{68}{243}\right)\) |
| \(\chi_{729}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{243}\right)\) | \(e\left(\frac{26}{243}\right)\) | \(e\left(\frac{56}{243}\right)\) | \(e\left(\frac{19}{243}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{34}{243}\right)\) | \(e\left(\frac{185}{243}\right)\) | \(e\left(\frac{32}{243}\right)\) | \(e\left(\frac{52}{243}\right)\) |
| \(\chi_{729}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{243}\right)\) | \(e\left(\frac{211}{243}\right)\) | \(e\left(\frac{118}{243}\right)\) | \(e\left(\frac{14}{243}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{89}{243}\right)\) | \(e\left(\frac{34}{243}\right)\) | \(e\left(\frac{241}{243}\right)\) | \(e\left(\frac{179}{243}\right)\) |
| \(\chi_{729}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{243}\right)\) | \(e\left(\frac{59}{243}\right)\) | \(e\left(\frac{71}{243}\right)\) | \(e\left(\frac{202}{243}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{208}{243}\right)\) | \(e\left(\frac{74}{243}\right)\) | \(e\left(\frac{110}{243}\right)\) | \(e\left(\frac{118}{243}\right)\) |
| \(\chi_{729}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{243}\right)\) | \(e\left(\frac{91}{243}\right)\) | \(e\left(\frac{196}{243}\right)\) | \(e\left(\frac{188}{243}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{119}{243}\right)\) | \(e\left(\frac{40}{243}\right)\) | \(e\left(\frac{112}{243}\right)\) | \(e\left(\frac{182}{243}\right)\) |
| \(\chi_{729}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{243}\right)\) | \(e\left(\frac{200}{243}\right)\) | \(e\left(\frac{113}{243}\right)\) | \(e\left(\frac{34}{243}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{112}{243}\right)\) | \(e\left(\frac{152}{243}\right)\) | \(e\left(\frac{134}{243}\right)\) | \(e\left(\frac{157}{243}\right)\) |
| \(\chi_{729}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{243}\right)\) | \(e\left(\frac{160}{243}\right)\) | \(e\left(\frac{139}{243}\right)\) | \(e\left(\frac{173}{243}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{41}{243}\right)\) | \(e\left(\frac{73}{243}\right)\) | \(e\left(\frac{10}{243}\right)\) | \(e\left(\frac{77}{243}\right)\) |
| \(\chi_{729}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{243}\right)\) | \(e\left(\frac{206}{243}\right)\) | \(e\left(\frac{182}{243}\right)\) | \(e\left(\frac{1}{243}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{232}{243}\right)\) | \(e\left(\frac{176}{243}\right)\) | \(e\left(\frac{104}{243}\right)\) | \(e\left(\frac{169}{243}\right)\) |
| \(\chi_{729}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{243}\right)\) | \(e\left(\frac{175}{243}\right)\) | \(e\left(\frac{190}{243}\right)\) | \(e\left(\frac{212}{243}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{98}{243}\right)\) | \(e\left(\frac{133}{243}\right)\) | \(e\left(\frac{178}{243}\right)\) | \(e\left(\frac{107}{243}\right)\) |
| \(\chi_{729}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{160}{243}\right)\) | \(e\left(\frac{77}{243}\right)\) | \(e\left(\frac{35}{243}\right)\) | \(e\left(\frac{103}{243}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{82}{243}\right)\) | \(e\left(\frac{146}{243}\right)\) | \(e\left(\frac{20}{243}\right)\) | \(e\left(\frac{154}{243}\right)\) |
| \(\chi_{729}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{243}\right)\) | \(e\left(\frac{136}{243}\right)\) | \(e\left(\frac{106}{243}\right)\) | \(e\left(\frac{62}{243}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{47}{243}\right)\) | \(e\left(\frac{220}{243}\right)\) | \(e\left(\frac{130}{243}\right)\) | \(e\left(\frac{29}{243}\right)\) |
| \(\chi_{729}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{243}\right)\) | \(e\left(\frac{56}{243}\right)\) | \(e\left(\frac{158}{243}\right)\) | \(e\left(\frac{97}{243}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{148}{243}\right)\) | \(e\left(\frac{62}{243}\right)\) | \(e\left(\frac{125}{243}\right)\) | \(e\left(\frac{112}{243}\right)\) |
| \(\chi_{729}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{243}\right)\) | \(e\left(\frac{43}{243}\right)\) | \(e\left(\frac{130}{243}\right)\) | \(e\left(\frac{209}{243}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{131}{243}\right)\) | \(e\left(\frac{91}{243}\right)\) | \(e\left(\frac{109}{243}\right)\) | \(e\left(\frac{86}{243}\right)\) |
| \(\chi_{729}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{243}\right)\) | \(e\left(\frac{143}{243}\right)\) | \(e\left(\frac{65}{243}\right)\) | \(e\left(\frac{226}{243}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{187}{243}\right)\) | \(e\left(\frac{167}{243}\right)\) | \(e\left(\frac{176}{243}\right)\) | \(e\left(\frac{43}{243}\right)\) |
| \(\chi_{729}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{243}\right)\) | \(e\left(\frac{139}{243}\right)\) | \(e\left(\frac{19}{243}\right)\) | \(e\left(\frac{167}{243}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{107}{243}\right)\) | \(e\left(\frac{232}{243}\right)\) | \(e\left(\frac{115}{243}\right)\) | \(e\left(\frac{35}{243}\right)\) |
| \(\chi_{729}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{243}\right)\) | \(e\left(\frac{95}{243}\right)\) | \(e\left(\frac{242}{243}\right)\) | \(e\left(\frac{4}{243}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{199}{243}\right)\) | \(e\left(\frac{218}{243}\right)\) | \(e\left(\frac{173}{243}\right)\) | \(e\left(\frac{190}{243}\right)\) |
| \(\chi_{729}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{212}{243}\right)\) | \(e\left(\frac{181}{243}\right)\) | \(e\left(\frac{16}{243}\right)\) | \(e\left(\frac{179}{243}\right)\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{218}{243}\right)\) | \(e\left(\frac{157}{243}\right)\) | \(e\left(\frac{148}{243}\right)\) | \(e\left(\frac{119}{243}\right)\) |
| \(\chi_{729}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{243}\right)\) | \(e\left(\frac{155}{243}\right)\) | \(e\left(\frac{203}{243}\right)\) | \(e\left(\frac{160}{243}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{184}{243}\right)\) | \(e\left(\frac{215}{243}\right)\) | \(e\left(\frac{116}{243}\right)\) | \(e\left(\frac{67}{243}\right)\) |
| \(\chi_{729}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{206}{243}\right)\) | \(e\left(\frac{169}{243}\right)\) | \(e\left(\frac{121}{243}\right)\) | \(e\left(\frac{2}{243}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{221}{243}\right)\) | \(e\left(\frac{109}{243}\right)\) | \(e\left(\frac{208}{243}\right)\) | \(e\left(\frac{95}{243}\right)\) |
| \(\chi_{729}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{243}\right)\) | \(e\left(\frac{80}{243}\right)\) | \(e\left(\frac{191}{243}\right)\) | \(e\left(\frac{208}{243}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{142}{243}\right)\) | \(e\left(\frac{158}{243}\right)\) | \(e\left(\frac{5}{243}\right)\) | \(e\left(\frac{160}{243}\right)\) |
| \(\chi_{729}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{243}\right)\) | \(e\left(\frac{103}{243}\right)\) | \(e\left(\frac{91}{243}\right)\) | \(e\left(\frac{122}{243}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{116}{243}\right)\) | \(e\left(\frac{88}{243}\right)\) | \(e\left(\frac{52}{243}\right)\) | \(e\left(\frac{206}{243}\right)\) |
| \(\chi_{729}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{178}{243}\right)\) | \(e\left(\frac{113}{243}\right)\) | \(e\left(\frac{206}{243}\right)\) | \(e\left(\frac{148}{243}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{73}{243}\right)\) | \(e\left(\frac{47}{243}\right)\) | \(e\left(\frac{83}{243}\right)\) | \(e\left(\frac{226}{243}\right)\) |
| \(\chi_{729}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{243}\right)\) | \(e\left(\frac{226}{243}\right)\) | \(e\left(\frac{169}{243}\right)\) | \(e\left(\frac{53}{243}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{146}{243}\right)\) | \(e\left(\frac{94}{243}\right)\) | \(e\left(\frac{166}{243}\right)\) | \(e\left(\frac{209}{243}\right)\) |
| \(\chi_{729}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{243}\right)\) | \(e\left(\frac{11}{243}\right)\) | \(e\left(\frac{5}{243}\right)\) | \(e\left(\frac{223}{243}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{220}{243}\right)\) | \(e\left(\frac{125}{243}\right)\) | \(e\left(\frac{107}{243}\right)\) | \(e\left(\frac{22}{243}\right)\) |