from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5054, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([0,29]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,5054))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5054\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(342\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 361.l | 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: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 342 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5054}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) |
| \(\chi_{5054}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{83}{114}\right)\) |
| \(\chi_{5054}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{341}{342}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) |
| \(\chi_{5054}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{125}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) |
| \(\chi_{5054}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{185}{342}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) |
| \(\chi_{5054}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{342}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{175}{342}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{29}{114}\right)\) |
| \(\chi_{5054}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{342}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{7}{342}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) |
| \(\chi_{5054}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{35}{342}\right)\) | \(e\left(\frac{185}{342}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{73}{114}\right)\) |
| \(\chi_{5054}(393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) |
| \(\chi_{5054}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{342}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{281}{342}\right)\) | \(e\left(\frac{215}{342}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{91}{114}\right)\) |
| \(\chi_{5054}(547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{193}{342}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{17}{114}\right)\) |
| \(\chi_{5054}(561,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{342}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{205}{342}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{47}{114}\right)\) |
| \(\chi_{5054}(603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{342}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{71}{342}\right)\) | \(e\left(\frac{131}{342}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) |
| \(\chi_{5054}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{342}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{79}{342}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) |
| \(\chi_{5054}(687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{259}{342}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{305}{342}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{31}{114}\right)\) |
| \(\chi_{5054}(743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{43}{114}\right)\) |
| \(\chi_{5054}(813,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{342}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{5}{114}\right)\) |
| \(\chi_{5054}(827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{342}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{61}{342}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{71}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{29}{114}\right)\) |
| \(\chi_{5054}(869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{31}{114}\right)\) |
| \(\chi_{5054}(925,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{342}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{71}{114}\right)\) |
| \(\chi_{5054}(953,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{41}{171}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{47}{342}\right)\) | \(e\left(\frac{53}{342}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) |
| \(\chi_{5054}(1009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{257}{342}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{67}{114}\right)\) |
| \(\chi_{5054}(1079,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{229}{342}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) |
| \(\chi_{5054}(1093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{49}{342}\right)\) | \(e\left(\frac{259}{342}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) |
| \(\chi_{5054}(1135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{143}{342}\right)\) | \(e\left(\frac{23}{342}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{67}{114}\right)\) |
| \(\chi_{5054}(1191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{342}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{101}{114}\right)\) |
| \(\chi_{5054}(1219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{101}{342}\right)\) | \(e\left(\frac{143}{342}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{25}{114}\right)\) |
| \(\chi_{5054}(1275,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{319}{342}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{101}{342}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{91}{114}\right)\) |
| \(\chi_{5054}(1359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{31}{342}\right)\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) |
| \(\chi_{5054}(1401,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{342}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{103}{114}\right)\) |
| \(\chi_{5054}(1457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{17}{114}\right)\) |