from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(379, base_ring=CyclotomicField(378))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,379))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(379\) | |
| Conductor: | \(379\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(189\) | 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: | $\Q(\zeta_{189})$ |
| Fixed field: | Number field defined by a degree 189 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{379}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{189}\right)\) | \(e\left(\frac{137}{189}\right)\) | \(e\left(\frac{2}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{155}{189}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{145}{189}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{379}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{189}\right)\) | \(e\left(\frac{58}{189}\right)\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{67}{189}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{116}{189}\right)\) | \(e\left(\frac{20}{189}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{379}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{189}\right)\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{4}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{121}{189}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{170}{189}\right)\) | \(e\left(\frac{101}{189}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{379}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{116}{189}\right)\) | \(e\left(\frac{16}{189}\right)\) | \(e\left(\frac{43}{189}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{25}{189}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{32}{189}\right)\) | \(e\left(\frac{188}{189}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{379}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{189}\right)\) | \(e\left(\frac{173}{189}\right)\) | \(e\left(\frac{146}{189}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{164}{189}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{157}{189}\right)\) | \(e\left(\frac{1}{189}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{379}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{189}\right)\) | \(e\left(\frac{157}{189}\right)\) | \(e\left(\frac{103}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{139}{189}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{125}{189}\right)\) | \(e\left(\frac{2}{189}\right)\) | \(e\left(\frac{17}{27}\right)\) |
| \(\chi_{379}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{189}\right)\) | \(e\left(\frac{2}{189}\right)\) | \(e\left(\frac{29}{189}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{74}{189}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{4}{189}\right)\) | \(e\left(\frac{118}{189}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{379}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{189}\right)\) | \(e\left(\frac{152}{189}\right)\) | \(e\left(\frac{125}{189}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{143}{189}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{115}{189}\right)\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{7}{27}\right)\) |
| \(\chi_{379}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{189}\right)\) | \(e\left(\frac{8}{189}\right)\) | \(e\left(\frac{116}{189}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{107}{189}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{16}{189}\right)\) | \(e\left(\frac{94}{189}\right)\) | \(e\left(\frac{16}{27}\right)\) |
| \(\chi_{379}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{189}\right)\) | \(e\left(\frac{94}{189}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{76}{189}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{188}{189}\right)\) | \(e\left(\frac{65}{189}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{379}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{189}\right)\) | \(e\left(\frac{67}{189}\right)\) | \(e\left(\frac{121}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{22}{189}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{134}{189}\right)\) | \(e\left(\frac{173}{189}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{379}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{189}\right)\) | \(e\left(\frac{61}{189}\right)\) | \(e\left(\frac{34}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{178}{189}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{122}{189}\right)\) | \(e\left(\frac{8}{189}\right)\) | \(e\left(\frac{14}{27}\right)\) |
| \(\chi_{379}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{189}\right)\) | \(e\left(\frac{50}{189}\right)\) | \(e\left(\frac{158}{189}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{149}{189}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{100}{189}\right)\) | \(e\left(\frac{115}{189}\right)\) | \(e\left(\frac{19}{27}\right)\) |
| \(\chi_{379}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{189}\right)\) | \(e\left(\frac{148}{189}\right)\) | \(e\left(\frac{67}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{184}{189}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{107}{189}\right)\) | \(e\left(\frac{38}{189}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{379}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{189}\right)\) | \(e\left(\frac{125}{189}\right)\) | \(e\left(\frac{17}{189}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{89}{189}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{61}{189}\right)\) | \(e\left(\frac{4}{189}\right)\) | \(e\left(\frac{7}{27}\right)\) |
| \(\chi_{379}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{189}\right)\) | \(e\left(\frac{121}{189}\right)\) | \(e\left(\frac{148}{189}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{130}{189}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{53}{189}\right)\) | \(e\left(\frac{146}{189}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{379}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{116}{189}\right)\) | \(e\left(\frac{170}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{134}{189}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{43}{189}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{16}{27}\right)\) |
| \(\chi_{379}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{110}{189}\right)\) | \(e\left(\frac{139}{189}\right)\) | \(e\left(\frac{31}{189}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{89}{189}\right)\) | \(e\left(\frac{74}{189}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{379}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{189}\right)\) | \(e\left(\frac{80}{189}\right)\) | \(e\left(\frac{26}{189}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{125}{189}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{160}{189}\right)\) | \(e\left(\frac{184}{189}\right)\) | \(e\left(\frac{25}{27}\right)\) |
| \(\chi_{379}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{188}{189}\right)\) | \(e\left(\frac{52}{189}\right)\) | \(e\left(\frac{187}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{34}{189}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{104}{189}\right)\) | \(e\left(\frac{44}{189}\right)\) | \(e\left(\frac{23}{27}\right)\) |
| \(\chi_{379}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{189}\right)\) | \(e\left(\frac{88}{189}\right)\) | \(e\left(\frac{142}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{43}{189}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{176}{189}\right)\) | \(e\left(\frac{89}{189}\right)\) | \(e\left(\frac{14}{27}\right)\) |
| \(\chi_{379}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{136}{189}\right)\) | \(e\left(\frac{110}{189}\right)\) | \(e\left(\frac{83}{189}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{101}{189}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{31}{189}\right)\) | \(e\left(\frac{64}{189}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{379}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{189}\right)\) | \(e\left(\frac{20}{189}\right)\) | \(e\left(\frac{101}{189}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{173}{189}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{46}{189}\right)\) | \(e\left(\frac{13}{27}\right)\) |
| \(\chi_{379}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{189}\right)\) | \(e\left(\frac{124}{189}\right)\) | \(e\left(\frac{97}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{52}{189}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{59}{189}\right)\) | \(e\left(\frac{134}{189}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{379}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{176}{189}\right)\) | \(e\left(\frac{109}{189}\right)\) | \(e\left(\frac{163}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{64}{189}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{29}{189}\right)\) | \(e\left(\frac{5}{189}\right)\) | \(e\left(\frac{2}{27}\right)\) |
| \(\chi_{379}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{158}{189}\right)\) | \(e\left(\frac{100}{189}\right)\) | \(e\left(\frac{127}{189}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{109}{189}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{11}{189}\right)\) | \(e\left(\frac{41}{189}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{379}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{189}\right)\) | \(e\left(\frac{4}{189}\right)\) | \(e\left(\frac{58}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{148}{189}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{8}{189}\right)\) | \(e\left(\frac{47}{189}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{379}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{189}\right)\) | \(e\left(\frac{146}{189}\right)\) | \(e\left(\frac{38}{189}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{110}{189}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{103}{189}\right)\) | \(e\left(\frac{109}{189}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{379}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{189}\right)\) | \(e\left(\frac{38}{189}\right)\) | \(e\left(\frac{173}{189}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{83}{189}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{76}{189}\right)\) | \(e\left(\frac{163}{189}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{379}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{189}\right)\) | \(e\left(\frac{19}{189}\right)\) | \(e\left(\frac{181}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{136}{189}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{38}{189}\right)\) | \(e\left(\frac{176}{189}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{379}(118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{170}{189}\right)\) | \(e\left(\frac{43}{189}\right)\) | \(e\left(\frac{151}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{79}{189}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{86}{189}\right)\) | \(e\left(\frac{80}{189}\right)\) | \(e\left(\frac{5}{27}\right)\) |