from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(2364))
M = H._module
chi = DirichletCharacter(H, M([299]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,4729))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4729\) | |
| Conductor: | \(4729\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2364\) | 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: | $\Q(\zeta_{2364})$ |
| Fixed field: | Number field defined by a degree 2364 polynomial (not computed) |
First 31 of 784 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4729}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{394}\right)\) | \(e\left(\frac{329}{394}\right)\) | \(e\left(\frac{87}{197}\right)\) | \(e\left(\frac{751}{1182}\right)\) | \(e\left(\frac{11}{197}\right)\) | \(e\left(\frac{787}{1182}\right)\) | \(e\left(\frac{261}{394}\right)\) | \(e\left(\frac{132}{197}\right)\) | \(e\left(\frac{506}{591}\right)\) | \(e\left(\frac{259}{788}\right)\) |
| \(\chi_{4729}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{394}\right)\) | \(e\left(\frac{339}{394}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{787}{1182}\right)\) | \(e\left(\frac{176}{197}\right)\) | \(e\left(\frac{181}{1182}\right)\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{142}{197}\right)\) | \(e\left(\frac{413}{591}\right)\) | \(e\left(\frac{401}{788}\right)\) |
| \(\chi_{4729}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{393}{394}\right)\) | \(e\left(\frac{277}{394}\right)\) | \(e\left(\frac{196}{197}\right)\) | \(e\left(\frac{91}{1182}\right)\) | \(e\left(\frac{138}{197}\right)\) | \(e\left(\frac{865}{1182}\right)\) | \(e\left(\frac{391}{394}\right)\) | \(e\left(\frac{80}{197}\right)\) | \(e\left(\frac{44}{591}\right)\) | \(e\left(\frac{545}{788}\right)\) |
| \(\chi_{4729}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{394}\right)\) | \(e\left(\frac{181}{394}\right)\) | \(e\left(\frac{79}{197}\right)\) | \(e\left(\frac{691}{1182}\right)\) | \(e\left(\frac{130}{197}\right)\) | \(e\left(\frac{1009}{1182}\right)\) | \(e\left(\frac{237}{394}\right)\) | \(e\left(\frac{181}{197}\right)\) | \(e\left(\frac{464}{591}\right)\) | \(e\left(\frac{285}{788}\right)\) |
| \(\chi_{4729}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{319}{394}\right)\) | \(e\left(\frac{287}{394}\right)\) | \(e\left(\frac{122}{197}\right)\) | \(e\left(\frac{127}{1182}\right)\) | \(e\left(\frac{106}{197}\right)\) | \(e\left(\frac{259}{1182}\right)\) | \(e\left(\frac{169}{394}\right)\) | \(e\left(\frac{90}{197}\right)\) | \(e\left(\frac{542}{591}\right)\) | \(e\left(\frac{687}{788}\right)\) |
| \(\chi_{4729}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{394}\right)\) | \(e\left(\frac{19}{394}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{817}{1182}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{661}{1182}\right)\) | \(e\left(\frac{51}{394}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{434}{591}\right)\) | \(e\left(\frac{585}{788}\right)\) |
| \(\chi_{4729}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{394}\right)\) | \(e\left(\frac{53}{394}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{703}{1182}\right)\) | \(e\left(\frac{185}{197}\right)\) | \(e\left(\frac{19}{1182}\right)\) | \(e\left(\frac{163}{394}\right)\) | \(e\left(\frac{53}{197}\right)\) | \(e\left(\frac{236}{591}\right)\) | \(e\left(\frac{595}{788}\right)\) |
| \(\chi_{4729}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{394}\right)\) | \(e\left(\frac{317}{394}\right)\) | \(e\left(\frac{97}{197}\right)\) | \(e\left(\frac{235}{1182}\right)\) | \(e\left(\frac{10}{197}\right)\) | \(e\left(\frac{805}{1182}\right)\) | \(e\left(\frac{291}{394}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{263}{591}\right)\) | \(e\left(\frac{325}{788}\right)\) |
| \(\chi_{4729}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{337}{394}\right)\) | \(e\left(\frac{29}{394}\right)\) | \(e\left(\frac{140}{197}\right)\) | \(e\left(\frac{853}{1182}\right)\) | \(e\left(\frac{183}{197}\right)\) | \(e\left(\frac{55}{1182}\right)\) | \(e\left(\frac{223}{394}\right)\) | \(e\left(\frac{29}{197}\right)\) | \(e\left(\frac{341}{591}\right)\) | \(e\left(\frac{727}{788}\right)\) |
| \(\chi_{4729}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{394}\right)\) | \(e\left(\frac{155}{394}\right)\) | \(e\left(\frac{35}{197}\right)\) | \(e\left(\frac{361}{1182}\right)\) | \(e\left(\frac{95}{197}\right)\) | \(e\left(\frac{457}{1182}\right)\) | \(e\left(\frac{105}{394}\right)\) | \(e\left(\frac{155}{197}\right)\) | \(e\left(\frac{233}{591}\right)\) | \(e\left(\frac{625}{788}\right)\) |
| \(\chi_{4729}(50,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{327}{394}\right)\) | \(e\left(\frac{41}{394}\right)\) | \(e\left(\frac{130}{197}\right)\) | \(e\left(\frac{581}{1182}\right)\) | \(e\left(\frac{184}{197}\right)\) | \(e\left(\frac{431}{1182}\right)\) | \(e\left(\frac{193}{394}\right)\) | \(e\left(\frac{41}{197}\right)\) | \(e\left(\frac{190}{591}\right)\) | \(e\left(\frac{661}{788}\right)\) |
| \(\chi_{4729}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{335}{394}\right)\) | \(e\left(\frac{189}{394}\right)\) | \(e\left(\frac{138}{197}\right)\) | \(e\left(\frac{247}{1182}\right)\) | \(e\left(\frac{65}{197}\right)\) | \(e\left(\frac{997}{1182}\right)\) | \(e\left(\frac{217}{394}\right)\) | \(e\left(\frac{189}{197}\right)\) | \(e\left(\frac{35}{591}\right)\) | \(e\left(\frac{635}{788}\right)\) |
| \(\chi_{4729}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{394}\right)\) | \(e\left(\frac{389}{394}\right)\) | \(e\left(\frac{37}{197}\right)\) | \(e\left(\frac{179}{1182}\right)\) | \(e\left(\frac{16}{197}\right)\) | \(e\left(\frac{1091}{1182}\right)\) | \(e\left(\frac{111}{394}\right)\) | \(e\left(\frac{192}{197}\right)\) | \(e\left(\frac{145}{591}\right)\) | \(e\left(\frac{717}{788}\right)\) |
| \(\chi_{4729}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{355}{394}\right)\) | \(e\left(\frac{165}{394}\right)\) | \(e\left(\frac{158}{197}\right)\) | \(e\left(\frac{397}{1182}\right)\) | \(e\left(\frac{63}{197}\right)\) | \(e\left(\frac{1033}{1182}\right)\) | \(e\left(\frac{277}{394}\right)\) | \(e\left(\frac{165}{197}\right)\) | \(e\left(\frac{140}{591}\right)\) | \(e\left(\frac{767}{788}\right)\) |
| \(\chi_{4729}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{394}\right)\) | \(e\left(\frac{339}{394}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{1181}{1182}\right)\) | \(e\left(\frac{176}{197}\right)\) | \(e\left(\frac{575}{1182}\right)\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{142}{197}\right)\) | \(e\left(\frac{19}{591}\right)\) | \(e\left(\frac{401}{788}\right)\) |
| \(\chi_{4729}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{394}\right)\) | \(e\left(\frac{51}{394}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{617}{1182}\right)\) | \(e\left(\frac{152}{197}\right)\) | \(e\left(\frac{1007}{1182}\right)\) | \(e\left(\frac{365}{394}\right)\) | \(e\left(\frac{51}{197}\right)\) | \(e\left(\frac{97}{591}\right)\) | \(e\left(\frac{15}{788}\right)\) |
| \(\chi_{4729}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{394}\right)\) | \(e\left(\frac{21}{394}\right)\) | \(e\left(\frac{81}{197}\right)\) | \(e\left(\frac{509}{1182}\right)\) | \(e\left(\frac{51}{197}\right)\) | \(e\left(\frac{461}{1182}\right)\) | \(e\left(\frac{243}{394}\right)\) | \(e\left(\frac{21}{197}\right)\) | \(e\left(\frac{376}{591}\right)\) | \(e\left(\frac{771}{788}\right)\) |
| \(\chi_{4729}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{394}\right)\) | \(e\left(\frac{345}{394}\right)\) | \(e\left(\frac{8}{197}\right)\) | \(e\left(\frac{1045}{1182}\right)\) | \(e\left(\frac{78}{197}\right)\) | \(e\left(\frac{763}{1182}\right)\) | \(e\left(\frac{221}{394}\right)\) | \(e\left(\frac{148}{197}\right)\) | \(e\left(\frac{239}{591}\right)\) | \(e\left(\frac{171}{788}\right)\) |
| \(\chi_{4729}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{394}\right)\) | \(e\left(\frac{101}{394}\right)\) | \(e\left(\frac{80}{197}\right)\) | \(e\left(\frac{403}{1182}\right)\) | \(e\left(\frac{189}{197}\right)\) | \(e\left(\frac{1129}{1182}\right)\) | \(e\left(\frac{43}{394}\right)\) | \(e\left(\frac{101}{197}\right)\) | \(e\left(\frac{26}{591}\right)\) | \(e\left(\frac{331}{788}\right)\) |
| \(\chi_{4729}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{345}{394}\right)\) | \(e\left(\frac{177}{394}\right)\) | \(e\left(\frac{148}{197}\right)\) | \(e\left(\frac{125}{1182}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{227}{1182}\right)\) | \(e\left(\frac{247}{394}\right)\) | \(e\left(\frac{177}{197}\right)\) | \(e\left(\frac{580}{591}\right)\) | \(e\left(\frac{701}{788}\right)\) |
| \(\chi_{4729}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{305}{394}\right)\) | \(e\left(\frac{225}{394}\right)\) | \(e\left(\frac{108}{197}\right)\) | \(e\left(\frac{613}{1182}\right)\) | \(e\left(\frac{68}{197}\right)\) | \(e\left(\frac{943}{1182}\right)\) | \(e\left(\frac{127}{394}\right)\) | \(e\left(\frac{28}{197}\right)\) | \(e\left(\frac{173}{591}\right)\) | \(e\left(\frac{43}{788}\right)\) |
| \(\chi_{4729}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{333}{394}\right)\) | \(e\left(\frac{349}{394}\right)\) | \(e\left(\frac{136}{197}\right)\) | \(e\left(\frac{35}{1182}\right)\) | \(e\left(\frac{144}{197}\right)\) | \(e\left(\frac{1151}{1182}\right)\) | \(e\left(\frac{211}{394}\right)\) | \(e\left(\frac{152}{197}\right)\) | \(e\left(\frac{517}{591}\right)\) | \(e\left(\frac{543}{788}\right)\) |
| \(\chi_{4729}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{394}\right)\) | \(e\left(\frac{211}{394}\right)\) | \(e\left(\frac{54}{197}\right)\) | \(e\left(\frac{11}{1182}\right)\) | \(e\left(\frac{34}{197}\right)\) | \(e\left(\frac{767}{1182}\right)\) | \(e\left(\frac{359}{394}\right)\) | \(e\left(\frac{14}{197}\right)\) | \(e\left(\frac{382}{591}\right)\) | \(e\left(\frac{711}{788}\right)\) |
| \(\chi_{4729}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{394}\right)\) | \(e\left(\frac{61}{394}\right)\) | \(e\left(\frac{179}{197}\right)\) | \(e\left(\frac{653}{1182}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{401}{1182}\right)\) | \(e\left(\frac{143}{394}\right)\) | \(e\left(\frac{61}{197}\right)\) | \(e\left(\frac{4}{591}\right)\) | \(e\left(\frac{157}{788}\right)\) |
| \(\chi_{4729}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{385}{394}\right)\) | \(e\left(\frac{129}{394}\right)\) | \(e\left(\frac{188}{197}\right)\) | \(e\left(\frac{31}{1182}\right)\) | \(e\left(\frac{60}{197}\right)\) | \(e\left(\frac{1087}{1182}\right)\) | \(e\left(\frac{367}{394}\right)\) | \(e\left(\frac{129}{197}\right)\) | \(e\left(\frac{2}{591}\right)\) | \(e\left(\frac{571}{788}\right)\) |
| \(\chi_{4729}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{394}\right)\) | \(e\left(\frac{187}{394}\right)\) | \(e\left(\frac{74}{197}\right)\) | \(e\left(\frac{161}{1182}\right)\) | \(e\left(\frac{32}{197}\right)\) | \(e\left(\frac{803}{1182}\right)\) | \(e\left(\frac{25}{394}\right)\) | \(e\left(\frac{187}{197}\right)\) | \(e\left(\frac{487}{591}\right)\) | \(e\left(\frac{55}{788}\right)\) |
| \(\chi_{4729}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{394}\right)\) | \(e\left(\frac{237}{394}\right)\) | \(e\left(\frac{98}{197}\right)\) | \(e\left(\frac{1129}{1182}\right)\) | \(e\left(\frac{69}{197}\right)\) | \(e\left(\frac{925}{1182}\right)\) | \(e\left(\frac{97}{394}\right)\) | \(e\left(\frac{40}{197}\right)\) | \(e\left(\frac{416}{591}\right)\) | \(e\left(\frac{371}{788}\right)\) |
| \(\chi_{4729}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{394}\right)\) | \(e\left(\frac{235}{394}\right)\) | \(e\left(\frac{34}{197}\right)\) | \(e\left(\frac{649}{1182}\right)\) | \(e\left(\frac{36}{197}\right)\) | \(e\left(\frac{337}{1182}\right)\) | \(e\left(\frac{299}{394}\right)\) | \(e\left(\frac{38}{197}\right)\) | \(e\left(\frac{80}{591}\right)\) | \(e\left(\frac{185}{788}\right)\) |
| \(\chi_{4729}(120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{323}{394}\right)\) | \(e\left(\frac{361}{394}\right)\) | \(e\left(\frac{126}{197}\right)\) | \(e\left(\frac{157}{1182}\right)\) | \(e\left(\frac{145}{197}\right)\) | \(e\left(\frac{739}{1182}\right)\) | \(e\left(\frac{181}{394}\right)\) | \(e\left(\frac{164}{197}\right)\) | \(e\left(\frac{563}{591}\right)\) | \(e\left(\frac{83}{788}\right)\) |
| \(\chi_{4729}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{394}\right)\) | \(e\left(\frac{221}{394}\right)\) | \(e\left(\frac{177}{197}\right)\) | \(e\left(\frac{47}{1182}\right)\) | \(e\left(\frac{2}{197}\right)\) | \(e\left(\frac{161}{1182}\right)\) | \(e\left(\frac{137}{394}\right)\) | \(e\left(\frac{24}{197}\right)\) | \(e\left(\frac{289}{591}\right)\) | \(e\left(\frac{65}{788}\right)\) |
| \(\chi_{4729}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{394}\right)\) | \(e\left(\frac{43}{394}\right)\) | \(e\left(\frac{194}{197}\right)\) | \(e\left(\frac{1061}{1182}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{1019}{1182}\right)\) | \(e\left(\frac{385}{394}\right)\) | \(e\left(\frac{43}{197}\right)\) | \(e\left(\frac{526}{591}\right)\) | \(e\left(\frac{59}{788}\right)\) |