from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([420,100,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,8036))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8036\) | |
| Conductor: | \(8036\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(840\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{840})$ |
| Fixed field: | Number field defined by a degree 840 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8036}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{283}{840}\right)\) | \(e\left(\frac{197}{280}\right)\) | \(e\left(\frac{279}{280}\right)\) | \(e\left(\frac{673}{840}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{1}{210}\right)\) |
| \(\chi_{8036}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{97}{840}\right)\) | \(e\left(\frac{23}{280}\right)\) | \(e\left(\frac{61}{280}\right)\) | \(e\left(\frac{667}{840}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{79}{210}\right)\) |
| \(\chi_{8036}(171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{557}{840}\right)\) | \(e\left(\frac{83}{280}\right)\) | \(e\left(\frac{1}{280}\right)\) | \(e\left(\frac{167}{840}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{209}{210}\right)\) |
| \(\chi_{8036}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{23}{840}\right)\) | \(e\left(\frac{17}{280}\right)\) | \(e\left(\frac{179}{280}\right)\) | \(e\left(\frac{773}{840}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{101}{210}\right)\) |
| \(\chi_{8036}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{121}{840}\right)\) | \(e\left(\frac{199}{280}\right)\) | \(e\left(\frac{53}{280}\right)\) | \(e\left(\frac{451}{840}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{157}{210}\right)\) |
| \(\chi_{8036}(311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{239}{840}\right)\) | \(e\left(\frac{201}{280}\right)\) | \(e\left(\frac{107}{280}\right)\) | \(e\left(\frac{509}{840}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{173}{210}\right)\) |
| \(\chi_{8036}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{209}{840}\right)\) | \(e\left(\frac{191}{280}\right)\) | \(e\left(\frac{117}{280}\right)\) | \(e\left(\frac{779}{840}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{23}{210}\right)\) |
| \(\chi_{8036}(395,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{611}{840}\right)\) | \(e\left(\frac{269}{280}\right)\) | \(e\left(\frac{263}{280}\right)\) | \(e\left(\frac{521}{840}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{17}{210}\right)\) |
| \(\chi_{8036}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{127}{420}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{661}{840}\right)\) | \(e\left(\frac{99}{280}\right)\) | \(e\left(\frac{153}{280}\right)\) | \(e\left(\frac{631}{840}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{127}{210}\right)\) |
| \(\chi_{8036}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{353}{840}\right)\) | \(e\left(\frac{127}{280}\right)\) | \(e\left(\frac{69}{280}\right)\) | \(e\left(\frac{323}{840}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{71}{210}\right)\) |
| \(\chi_{8036}(507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{71}{840}\right)\) | \(e\left(\frac{89}{280}\right)\) | \(e\left(\frac{163}{280}\right)\) | \(e\left(\frac{341}{840}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{47}{210}\right)\) |
| \(\chi_{8036}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{389}{840}\right)\) | \(e\left(\frac{251}{280}\right)\) | \(e\left(\frac{57}{280}\right)\) | \(e\left(\frac{839}{840}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{83}{210}\right)\) |
| \(\chi_{8036}(591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{779}{840}\right)\) | \(e\left(\frac{101}{280}\right)\) | \(e\left(\frac{207}{280}\right)\) | \(e\left(\frac{689}{840}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{143}{210}\right)\) |
| \(\chi_{8036}(663,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{517}{840}\right)\) | \(e\left(\frac{163}{280}\right)\) | \(e\left(\frac{201}{280}\right)\) | \(e\left(\frac{247}{840}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{79}{210}\right)\) |
| \(\chi_{8036}(675,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{227}{840}\right)\) | \(e\left(\frac{253}{280}\right)\) | \(e\left(\frac{111}{280}\right)\) | \(e\left(\frac{617}{840}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{29}{210}\right)\) |
| \(\chi_{8036}(691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{583}{840}\right)\) | \(e\left(\frac{17}{280}\right)\) | \(e\left(\frac{179}{280}\right)\) | \(e\left(\frac{493}{840}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{31}{210}\right)\) |
| \(\chi_{8036}(703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{323}{840}\right)\) | \(e\left(\frac{117}{280}\right)\) | \(e\left(\frac{79}{280}\right)\) | \(e\left(\frac{593}{840}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{131}{210}\right)\) |
| \(\chi_{8036}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{607}{840}\right)\) | \(e\left(\frac{193}{280}\right)\) | \(e\left(\frac{171}{280}\right)\) | \(e\left(\frac{277}{840}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{109}{210}\right)\) |
| \(\chi_{8036}(731,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{377}{840}\right)\) | \(e\left(\frac{23}{280}\right)\) | \(e\left(\frac{61}{280}\right)\) | \(e\left(\frac{107}{840}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{149}{210}\right)\) |
| \(\chi_{8036}(831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{409}{840}\right)\) | \(e\left(\frac{71}{280}\right)\) | \(e\left(\frac{237}{280}\right)\) | \(e\left(\frac{379}{840}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{43}{210}\right)\) |
| \(\chi_{8036}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{157}{420}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{331}{840}\right)\) | \(e\left(\frac{269}{280}\right)\) | \(e\left(\frac{263}{280}\right)\) | \(e\left(\frac{241}{840}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{157}{210}\right)\) |
| \(\chi_{8036}(915,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{733}{840}\right)\) | \(e\left(\frac{67}{280}\right)\) | \(e\left(\frac{129}{280}\right)\) | \(e\left(\frac{823}{840}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{151}{210}\right)\) |
| \(\chi_{8036}(955,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{167}{420}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{641}{840}\right)\) | \(e\left(\frac{279}{280}\right)\) | \(e\left(\frac{253}{280}\right)\) | \(e\left(\frac{251}{840}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{167}{210}\right)\) |
| \(\chi_{8036}(971,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{193}{840}\right)\) | \(e\left(\frac{167}{280}\right)\) | \(e\left(\frac{29}{280}\right)\) | \(e\left(\frac{643}{840}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{181}{210}\right)\) |
| \(\chi_{8036}(1055,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{349}{840}\right)\) | \(e\left(\frac{51}{280}\right)\) | \(e\left(\frac{257}{280}\right)\) | \(e\left(\frac{79}{840}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{163}{210}\right)\) |
| \(\chi_{8036}(1083,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{499}{840}\right)\) | \(e\left(\frac{101}{280}\right)\) | \(e\left(\frac{207}{280}\right)\) | \(e\left(\frac{409}{840}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{73}{210}\right)\) |
| \(\chi_{8036}(1095,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{701}{840}\right)\) | \(e\left(\frac{19}{280}\right)\) | \(e\left(\frac{233}{280}\right)\) | \(e\left(\frac{551}{840}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{47}{210}\right)\) |
| \(\chi_{8036}(1167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{67}{840}\right)\) | \(e\left(\frac{13}{280}\right)\) | \(e\left(\frac{71}{280}\right)\) | \(e\left(\frac{97}{840}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{139}{210}\right)\) |
| \(\chi_{8036}(1223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{577}{840}\right)\) | \(e\left(\frac{183}{280}\right)\) | \(e\left(\frac{181}{280}\right)\) | \(e\left(\frac{547}{840}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{169}{210}\right)\) |
| \(\chi_{8036}(1319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{797}{840}\right)\) | \(e\left(\frac{163}{280}\right)\) | \(e\left(\frac{201}{280}\right)\) | \(e\left(\frac{527}{840}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{149}{210}\right)\) |
| \(\chi_{8036}(1347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{263}{840}\right)\) | \(e\left(\frac{97}{280}\right)\) | \(e\left(\frac{99}{280}\right)\) | \(e\left(\frac{293}{840}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{41}{210}\right)\) |