from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1337, base_ring=CyclotomicField(570))
M = H._module
chi = DirichletCharacter(H, M([190,132]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1337))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1337\) | |
| Conductor: | \(1337\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(285\) | 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: | $\Q(\zeta_{285})$ |
| Fixed field: | Number field defined by a degree 285 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1337}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{244}{285}\right)\) | \(e\left(\frac{56}{285}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{112}{285}\right)\) | \(e\left(\frac{29}{285}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{259}{285}\right)\) |
| \(\chi_{1337}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{112}{285}\right)\) | \(e\left(\frac{121}{285}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{95}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{233}{285}\right)\) |
| \(\chi_{1337}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{285}\right)\) | \(e\left(\frac{278}{285}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{32}{285}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{217}{285}\right)\) |
| \(\chi_{1337}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{285}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{242}{285}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{26}{95}\right)\) | \(e\left(\frac{163}{285}\right)\) | \(e\left(\frac{116}{285}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{181}{285}\right)\) |
| \(\chi_{1337}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{285}\right)\) | \(e\left(\frac{49}{285}\right)\) | \(e\left(\frac{142}{285}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{98}{285}\right)\) | \(e\left(\frac{61}{285}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{191}{285}\right)\) |
| \(\chi_{1337}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{285}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{113}{285}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{82}{285}\right)\) | \(e\left(\frac{179}{285}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{154}{285}\right)\) |
| \(\chi_{1337}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{158}{285}\right)\) | \(e\left(\frac{97}{285}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{63}{95}\right)\) | \(e\left(\frac{194}{285}\right)\) | \(e\left(\frac{208}{285}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{128}{285}\right)\) |
| \(\chi_{1337}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{281}{285}\right)\) | \(e\left(\frac{128}{285}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{277}{285}\right)\) | \(e\left(\frac{59}{285}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{124}{285}\right)\) |
| \(\chi_{1337}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{285}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{137}{285}\right)\) | \(e\left(\frac{94}{285}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{14}{285}\right)\) |
| \(\chi_{1337}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{68}{285}\right)\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{136}{285}\right)\) | \(e\left(\frac{137}{285}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{172}{285}\right)\) |
| \(\chi_{1337}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{285}\right)\) | \(e\left(\frac{67}{285}\right)\) | \(e\left(\frac{136}{285}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{68}{95}\right)\) | \(e\left(\frac{134}{285}\right)\) | \(e\left(\frac{223}{285}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{203}{285}\right)\) |
| \(\chi_{1337}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{274}{285}\right)\) | \(e\left(\frac{161}{285}\right)\) | \(e\left(\frac{263}{285}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{84}{95}\right)\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{139}{285}\right)\) |
| \(\chi_{1337}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{285}\right)\) | \(e\left(\frac{53}{285}\right)\) | \(e\left(\frac{14}{285}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{106}{285}\right)\) | \(e\left(\frac{2}{285}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{67}{285}\right)\) |
| \(\chi_{1337}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{163}{285}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{257}{285}\right)\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{149}{285}\right)\) |
| \(\chi_{1337}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{202}{285}\right)\) | \(e\left(\frac{23}{285}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{12}{95}\right)\) | \(e\left(\frac{46}{285}\right)\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{142}{285}\right)\) |
| \(\chi_{1337}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{32}{285}\right)\) | \(e\left(\frac{116}{285}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{58}{95}\right)\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{98}{285}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{148}{285}\right)\) |
| \(\chi_{1337}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{285}\right)\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{46}{285}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{95}\right)\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{98}{285}\right)\) |
| \(\chi_{1337}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{285}\right)\) | \(e\left(\frac{107}{285}\right)\) | \(e\left(\frac{281}{285}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{214}{285}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{103}{285}\right)\) |
| \(\chi_{1337}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{285}\right)\) | \(e\left(\frac{124}{285}\right)\) | \(e\left(\frac{22}{285}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{248}{285}\right)\) | \(e\left(\frac{166}{285}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{146}{285}\right)\) |
| \(\chi_{1337}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{191}{285}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{48}{95}\right)\) | \(e\left(\frac{184}{285}\right)\) | \(e\left(\frac{68}{285}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{283}{285}\right)\) |
| \(\chi_{1337}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{285}\right)\) | \(e\left(\frac{217}{285}\right)\) | \(e\left(\frac{181}{285}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{149}{285}\right)\) | \(e\left(\frac{148}{285}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{113}{285}\right)\) |
| \(\chi_{1337}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{285}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{182}{285}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{91}{95}\right)\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{26}{285}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{16}{285}\right)\) |
| \(\chi_{1337}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{285}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{197}{285}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{51}{95}\right)\) | \(e\left(\frac{148}{285}\right)\) | \(e\left(\frac{191}{285}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{271}{285}\right)\) |
| \(\chi_{1337}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{285}\right)\) | \(e\left(\frac{109}{285}\right)\) | \(e\left(\frac{217}{285}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{61}{95}\right)\) | \(e\left(\frac{218}{285}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{41}{285}\right)\) |
| \(\chi_{1337}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{184}{285}\right)\) | \(e\left(\frac{131}{285}\right)\) | \(e\left(\frac{83}{285}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{89}{95}\right)\) | \(e\left(\frac{262}{285}\right)\) | \(e\left(\frac{134}{285}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{214}{285}\right)\) |
| \(\chi_{1337}(170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{158}{285}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{95}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{232}{285}\right)\) |
| \(\chi_{1337}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{285}\right)\) | \(e\left(\frac{79}{285}\right)\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{66}{95}\right)\) | \(e\left(\frac{158}{285}\right)\) | \(e\left(\frac{46}{285}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{116}{285}\right)\) |
| \(\chi_{1337}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{285}\right)\) | \(e\left(\frac{151}{285}\right)\) | \(e\left(\frac{13}{285}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{124}{285}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{164}{285}\right)\) |
| \(\chi_{1337}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{34}{285}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{176}{285}\right)\) | \(e\left(\frac{127}{285}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{122}{285}\right)\) |
| \(\chi_{1337}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{285}\right)\) | \(e\left(\frac{34}{285}\right)\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{26}{95}\right)\) | \(e\left(\frac{68}{285}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{86}{285}\right)\) |
| \(\chi_{1337}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{136}{285}\right)\) | \(e\left(\frac{208}{285}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{272}{285}\right)\) | \(e\left(\frac{274}{285}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{59}{285}\right)\) |