from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4067, base_ring=CyclotomicField(246))
M = H._module
chi = DirichletCharacter(H, M([164,189]))
chi.galois_orbit()
[g,chi] = znchar(Mod(18,4067))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4067\) | |
| Conductor: | \(581\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(246\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 581.o | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{123})$ |
| Fixed field: | Number field defined by a degree 246 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4067}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{246}\right)\) | \(e\left(\frac{121}{123}\right)\) | \(e\left(\frac{25}{123}\right)\) | \(e\left(\frac{19}{246}\right)\) | \(e\left(\frac{7}{82}\right)\) | \(e\left(\frac{25}{82}\right)\) | \(e\left(\frac{119}{123}\right)\) | \(e\left(\frac{22}{123}\right)\) | \(e\left(\frac{13}{123}\right)\) | \(e\left(\frac{23}{123}\right)\) |
| \(\chi_{4067}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{246}\right)\) | \(e\left(\frac{22}{123}\right)\) | \(e\left(\frac{94}{123}\right)\) | \(e\left(\frac{37}{246}\right)\) | \(e\left(\frac{5}{82}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{44}{123}\right)\) | \(e\left(\frac{4}{123}\right)\) | \(e\left(\frac{103}{123}\right)\) | \(e\left(\frac{116}{123}\right)\) |
| \(\chi_{4067}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{246}\right)\) | \(e\left(\frac{11}{123}\right)\) | \(e\left(\frac{47}{123}\right)\) | \(e\left(\frac{203}{246}\right)\) | \(e\left(\frac{23}{82}\right)\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{22}{123}\right)\) | \(e\left(\frac{2}{123}\right)\) | \(e\left(\frac{113}{123}\right)\) | \(e\left(\frac{58}{123}\right)\) |
| \(\chi_{4067}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{246}\right)\) | \(e\left(\frac{59}{123}\right)\) | \(e\left(\frac{62}{123}\right)\) | \(e\left(\frac{239}{246}\right)\) | \(e\left(\frac{19}{82}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{118}{123}\right)\) | \(e\left(\frac{89}{123}\right)\) | \(e\left(\frac{47}{123}\right)\) | \(e\left(\frac{121}{123}\right)\) |
| \(\chi_{4067}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{246}\right)\) | \(e\left(\frac{2}{123}\right)\) | \(e\left(\frac{98}{123}\right)\) | \(e\left(\frac{227}{246}\right)\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{57}{82}\right)\) | \(e\left(\frac{4}{123}\right)\) | \(e\left(\frac{101}{123}\right)\) | \(e\left(\frac{110}{123}\right)\) | \(e\left(\frac{100}{123}\right)\) |
| \(\chi_{4067}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{246}\right)\) | \(e\left(\frac{70}{123}\right)\) | \(e\left(\frac{109}{123}\right)\) | \(e\left(\frac{73}{246}\right)\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{17}{123}\right)\) | \(e\left(\frac{91}{123}\right)\) | \(e\left(\frac{37}{123}\right)\) | \(e\left(\frac{56}{123}\right)\) |
| \(\chi_{4067}(471,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{246}\right)\) | \(e\left(\frac{122}{123}\right)\) | \(e\left(\frac{74}{123}\right)\) | \(e\left(\frac{71}{246}\right)\) | \(e\left(\frac{65}{82}\right)\) | \(e\left(\frac{33}{82}\right)\) | \(e\left(\frac{121}{123}\right)\) | \(e\left(\frac{11}{123}\right)\) | \(e\left(\frac{68}{123}\right)\) | \(e\left(\frac{73}{123}\right)\) |
| \(\chi_{4067}(520,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{246}\right)\) | \(e\left(\frac{35}{123}\right)\) | \(e\left(\frac{116}{123}\right)\) | \(e\left(\frac{221}{246}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{70}{123}\right)\) | \(e\left(\frac{107}{123}\right)\) | \(e\left(\frac{80}{123}\right)\) | \(e\left(\frac{28}{123}\right)\) |
| \(\chi_{4067}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{246}\right)\) | \(e\left(\frac{38}{123}\right)\) | \(e\left(\frac{17}{123}\right)\) | \(e\left(\frac{131}{246}\right)\) | \(e\left(\frac{31}{82}\right)\) | \(e\left(\frac{17}{82}\right)\) | \(e\left(\frac{76}{123}\right)\) | \(e\left(\frac{74}{123}\right)\) | \(e\left(\frac{122}{123}\right)\) | \(e\left(\frac{55}{123}\right)\) |
| \(\chi_{4067}(655,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{246}\right)\) | \(e\left(\frac{13}{123}\right)\) | \(e\left(\frac{22}{123}\right)\) | \(e\left(\frac{61}{246}\right)\) | \(e\left(\frac{57}{82}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{26}{123}\right)\) | \(e\left(\frac{103}{123}\right)\) | \(e\left(\frac{100}{123}\right)\) | \(e\left(\frac{35}{123}\right)\) |
| \(\chi_{4067}(716,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{246}\right)\) | \(e\left(\frac{86}{123}\right)\) | \(e\left(\frac{32}{123}\right)\) | \(e\left(\frac{167}{246}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{49}{123}\right)\) | \(e\left(\frac{38}{123}\right)\) | \(e\left(\frac{56}{123}\right)\) | \(e\left(\frac{118}{123}\right)\) |
| \(\chi_{4067}(753,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{246}\right)\) | \(e\left(\frac{94}{123}\right)\) | \(e\left(\frac{55}{123}\right)\) | \(e\left(\frac{91}{246}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{55}{82}\right)\) | \(e\left(\frac{65}{123}\right)\) | \(e\left(\frac{73}{123}\right)\) | \(e\left(\frac{4}{123}\right)\) | \(e\left(\frac{26}{123}\right)\) |
| \(\chi_{4067}(765,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{246}\right)\) | \(e\left(\frac{80}{123}\right)\) | \(e\left(\frac{107}{123}\right)\) | \(e\left(\frac{101}{246}\right)\) | \(e\left(\frac{7}{82}\right)\) | \(e\left(\frac{25}{82}\right)\) | \(e\left(\frac{37}{123}\right)\) | \(e\left(\frac{104}{123}\right)\) | \(e\left(\frac{95}{123}\right)\) | \(e\left(\frac{64}{123}\right)\) |
| \(\chi_{4067}(802,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{246}\right)\) | \(e\left(\frac{55}{123}\right)\) | \(e\left(\frac{112}{123}\right)\) | \(e\left(\frac{31}{246}\right)\) | \(e\left(\frac{33}{82}\right)\) | \(e\left(\frac{71}{82}\right)\) | \(e\left(\frac{110}{123}\right)\) | \(e\left(\frac{10}{123}\right)\) | \(e\left(\frac{73}{123}\right)\) | \(e\left(\frac{44}{123}\right)\) |
| \(\chi_{4067}(814,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{246}\right)\) | \(e\left(\frac{104}{123}\right)\) | \(e\left(\frac{53}{123}\right)\) | \(e\left(\frac{119}{246}\right)\) | \(e\left(\frac{5}{82}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{85}{123}\right)\) | \(e\left(\frac{86}{123}\right)\) | \(e\left(\frac{62}{123}\right)\) | \(e\left(\frac{34}{123}\right)\) |
| \(\chi_{4067}(998,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{246}\right)\) | \(e\left(\frac{67}{123}\right)\) | \(e\left(\frac{85}{123}\right)\) | \(e\left(\frac{163}{246}\right)\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{11}{123}\right)\) | \(e\left(\frac{1}{123}\right)\) | \(e\left(\frac{118}{123}\right)\) | \(e\left(\frac{29}{123}\right)\) |
| \(\chi_{4067}(1010,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{246}\right)\) | \(e\left(\frac{29}{123}\right)\) | \(e\left(\frac{68}{123}\right)\) | \(e\left(\frac{155}{246}\right)\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{58}{123}\right)\) | \(e\left(\frac{50}{123}\right)\) | \(e\left(\frac{119}{123}\right)\) | \(e\left(\frac{97}{123}\right)\) |
| \(\chi_{4067}(1145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{246}\right)\) | \(e\left(\frac{103}{123}\right)\) | \(e\left(\frac{4}{123}\right)\) | \(e\left(\frac{67}{246}\right)\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{83}{123}\right)\) | \(e\left(\frac{97}{123}\right)\) | \(e\left(\frac{7}{123}\right)\) | \(e\left(\frac{107}{123}\right)\) |
| \(\chi_{4067}(1194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{246}\right)\) | \(e\left(\frac{7}{123}\right)\) | \(e\left(\frac{97}{123}\right)\) | \(e\left(\frac{241}{246}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{15}{82}\right)\) | \(e\left(\frac{14}{123}\right)\) | \(e\left(\frac{46}{123}\right)\) | \(e\left(\frac{16}{123}\right)\) | \(e\left(\frac{104}{123}\right)\) |
| \(\chi_{4067}(1292,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{246}\right)\) | \(e\left(\frac{106}{123}\right)\) | \(e\left(\frac{28}{123}\right)\) | \(e\left(\frac{223}{246}\right)\) | \(e\left(\frac{39}{82}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{89}{123}\right)\) | \(e\left(\frac{64}{123}\right)\) | \(e\left(\frac{49}{123}\right)\) | \(e\left(\frac{11}{123}\right)\) |
| \(\chi_{4067}(1341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{246}\right)\) | \(e\left(\frac{34}{123}\right)\) | \(e\left(\frac{67}{123}\right)\) | \(e\left(\frac{169}{246}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{67}{82}\right)\) | \(e\left(\frac{68}{123}\right)\) | \(e\left(\frac{118}{123}\right)\) | \(e\left(\frac{25}{123}\right)\) | \(e\left(\frac{101}{123}\right)\) |
| \(\chi_{4067}(1390,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{246}\right)\) | \(e\left(\frac{112}{123}\right)\) | \(e\left(\frac{76}{123}\right)\) | \(e\left(\frac{43}{246}\right)\) | \(e\left(\frac{59}{82}\right)\) | \(e\left(\frac{35}{82}\right)\) | \(e\left(\frac{101}{123}\right)\) | \(e\left(\frac{121}{123}\right)\) | \(e\left(\frac{10}{123}\right)\) | \(e\left(\frac{65}{123}\right)\) |
| \(\chi_{4067}(1402,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{246}\right)\) | \(e\left(\frac{95}{123}\right)\) | \(e\left(\frac{104}{123}\right)\) | \(e\left(\frac{143}{246}\right)\) | \(e\left(\frac{57}{82}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{67}{123}\right)\) | \(e\left(\frac{62}{123}\right)\) | \(e\left(\frac{59}{123}\right)\) | \(e\left(\frac{76}{123}\right)\) |
| \(\chi_{4067}(1500,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{246}\right)\) | \(e\left(\frac{53}{123}\right)\) | \(e\left(\frac{14}{123}\right)\) | \(e\left(\frac{173}{246}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{55}{82}\right)\) | \(e\left(\frac{106}{123}\right)\) | \(e\left(\frac{32}{123}\right)\) | \(e\left(\frac{86}{123}\right)\) | \(e\left(\frac{67}{123}\right)\) |
| \(\chi_{4067}(1537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{246}\right)\) | \(e\left(\frac{1}{123}\right)\) | \(e\left(\frac{49}{123}\right)\) | \(e\left(\frac{175}{246}\right)\) | \(e\left(\frac{17}{82}\right)\) | \(e\left(\frac{49}{82}\right)\) | \(e\left(\frac{2}{123}\right)\) | \(e\left(\frac{112}{123}\right)\) | \(e\left(\frac{55}{123}\right)\) | \(e\left(\frac{50}{123}\right)\) |
| \(\chi_{4067}(1549,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{246}\right)\) | \(e\left(\frac{14}{123}\right)\) | \(e\left(\frac{71}{123}\right)\) | \(e\left(\frac{113}{246}\right)\) | \(e\left(\frac{33}{82}\right)\) | \(e\left(\frac{71}{82}\right)\) | \(e\left(\frac{28}{123}\right)\) | \(e\left(\frac{92}{123}\right)\) | \(e\left(\frac{32}{123}\right)\) | \(e\left(\frac{85}{123}\right)\) |
| \(\chi_{4067}(1635,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{246}\right)\) | \(e\left(\frac{10}{123}\right)\) | \(e\left(\frac{121}{123}\right)\) | \(e\left(\frac{151}{246}\right)\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{39}{82}\right)\) | \(e\left(\frac{20}{123}\right)\) | \(e\left(\frac{13}{123}\right)\) | \(e\left(\frac{58}{123}\right)\) | \(e\left(\frac{8}{123}\right)\) |
| \(\chi_{4067}(1684,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{246}\right)\) | \(e\left(\frac{64}{123}\right)\) | \(e\left(\frac{61}{123}\right)\) | \(e\left(\frac{7}{246}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{5}{123}\right)\) | \(e\left(\frac{34}{123}\right)\) | \(e\left(\frac{76}{123}\right)\) | \(e\left(\frac{2}{123}\right)\) |
| \(\chi_{4067}(1733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{246}\right)\) | \(e\left(\frac{31}{123}\right)\) | \(e\left(\frac{43}{123}\right)\) | \(e\left(\frac{13}{246}\right)\) | \(e\left(\frac{35}{82}\right)\) | \(e\left(\frac{43}{82}\right)\) | \(e\left(\frac{62}{123}\right)\) | \(e\left(\frac{28}{123}\right)\) | \(e\left(\frac{106}{123}\right)\) | \(e\left(\frac{74}{123}\right)\) |
| \(\chi_{4067}(1745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{246}\right)\) | \(e\left(\frac{26}{123}\right)\) | \(e\left(\frac{44}{123}\right)\) | \(e\left(\frac{245}{246}\right)\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{52}{123}\right)\) | \(e\left(\frac{83}{123}\right)\) | \(e\left(\frac{77}{123}\right)\) | \(e\left(\frac{70}{123}\right)\) |
| \(\chi_{4067}(1782,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{246}\right)\) | \(e\left(\frac{61}{123}\right)\) | \(e\left(\frac{37}{123}\right)\) | \(e\left(\frac{97}{246}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{122}{123}\right)\) | \(e\left(\frac{67}{123}\right)\) | \(e\left(\frac{34}{123}\right)\) | \(e\left(\frac{98}{123}\right)\) |