from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([234,416,423]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,6042))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6042\) | |
| Conductor: | \(3021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(468\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3021.ct | 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: | $\Q(\zeta_{468})$ |
| Fixed field: | Number field defined by a degree 468 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6042}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{468}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{16}{117}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{95}{234}\right)\) | \(e\left(\frac{22}{117}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{89}{468}\right)\) |
| \(\chi_{6042}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{468}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{28}{117}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{443}{468}\right)\) |
| \(\chi_{6042}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{433}{468}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{37}{117}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{199}{234}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{115}{468}\right)\) |
| \(\chi_{6042}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{468}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{112}{117}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{307}{468}\right)\) |
| \(\chi_{6042}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{468}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{80}{117}\right)\) | \(e\left(\frac{16}{117}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{110}{117}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{211}{468}\right)\) |
| \(\chi_{6042}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{468}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{89}{117}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{259}{468}\right)\) |
| \(\chi_{6042}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{347}{468}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{113}{234}\right)\) | \(e\left(\frac{4}{117}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{197}{468}\right)\) |
| \(\chi_{6042}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{468}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{34}{117}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{329}{468}\right)\) |
| \(\chi_{6042}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{468}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{77}{117}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{19}{468}\right)\) |
| \(\chi_{6042}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{468}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{67}{117}\right)\) | \(e\left(\frac{107}{117}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{215}{234}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{29}{468}\right)\) |
| \(\chi_{6042}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{468}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{239}{468}\right)\) |
| \(\chi_{6042}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{468}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{44}{117}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{115}{234}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{157}{468}\right)\) |
| \(\chi_{6042}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{468}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{7}{468}\right)\) |
| \(\chi_{6042}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{468}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{94}{117}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{85}{234}\right)\) | \(e\left(\frac{32}{117}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{55}{468}\right)\) |
| \(\chi_{6042}(491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{468}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{4}{117}\right)\) | \(e\left(\frac{71}{117}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{461}{468}\right)\) |
| \(\chi_{6042}(503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{468}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{89}{117}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{20}{117}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{283}{468}\right)\) |
| \(\chi_{6042}(557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{468}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{10}{117}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{205}{468}\right)\) |
| \(\chi_{6042}(575,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{437}{468}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{35}{468}\right)\) |
| \(\chi_{6042}(605,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{468}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{40}{117}\right)\) | \(e\left(\frac{8}{117}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{179}{234}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{281}{468}\right)\) |
| \(\chi_{6042}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{468}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{234}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{85}{468}\right)\) |
| \(\chi_{6042}(671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{468}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{1}{117}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{80}{117}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{79}{468}\right)\) |
| \(\chi_{6042}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{468}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{5}{468}\right)\) |
| \(\chi_{6042}(707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{427}{468}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{40}{117}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{193}{234}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{1}{468}\right)\) |
| \(\chi_{6042}(803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{468}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{269}{468}\right)\) |
| \(\chi_{6042}(815,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{468}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{56}{117}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{34}{117}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{95}{468}\right)\) |
| \(\chi_{6042}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{409}{468}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{11}{117}\right)\) | \(e\left(\frac{49}{117}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{59}{117}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{127}{468}\right)\) |
| \(\chi_{6042}(845,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{468}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{8}{117}\right)\) | \(e\left(\frac{25}{117}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{11}{117}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{337}{468}\right)\) |
| \(\chi_{6042}(899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{468}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{71}{117}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{133}{468}\right)\) |
| \(\chi_{6042}(935,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{468}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{85}{117}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{14}{117}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{397}{468}\right)\) |
| \(\chi_{6042}(959,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{468}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{107}{117}\right)\) | \(e\left(\frac{115}{117}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{43}{234}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{193}{468}\right)\) |
| \(\chi_{6042}(1127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{468}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{95}{117}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{234}\right)\) | \(e\left(\frac{116}{117}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{97}{468}\right)\) |