from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3971, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([171,233]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,3971))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3971\) | |
| Conductor: | \(3971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(342\) | 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: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 342 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3971}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{301}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{41}{171}\right)\) | \(e\left(\frac{7}{114}\right)\) |
| \(\chi_{3971}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{275}{342}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{17}{114}\right)\) |
| \(\chi_{3971}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{7}{114}\right)\) |
| \(\chi_{3971}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) |
| \(\chi_{3971}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{91}{342}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{89}{114}\right)\) |
| \(\chi_{3971}(186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{289}{342}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{73}{114}\right)\) |
| \(\chi_{3971}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{55}{114}\right)\) |
| \(\chi_{3971}(230,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{29}{114}\right)\) |
| \(\chi_{3971}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{205}{342}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{79}{114}\right)\) |
| \(\chi_{3971}(318,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{217}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) |
| \(\chi_{3971}(395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{89}{342}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{67}{114}\right)\) |
| \(\chi_{3971}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{103}{114}\right)\) |
| \(\chi_{3971}(439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) |
| \(\chi_{3971}(450,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) |
| \(\chi_{3971}(516,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) |
| \(\chi_{3971}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) |
| \(\chi_{3971}(604,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{61}{114}\right)\) |
| \(\chi_{3971}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) |
| \(\chi_{3971}(648,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{53}{114}\right)\) |
| \(\chi_{3971}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{155}{342}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{241}{342}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) |
| \(\chi_{3971}(725,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{35}{114}\right)\) |
| \(\chi_{3971}(736,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{127}{342}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{29}{114}\right)\) |
| \(\chi_{3971}(813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{233}{342}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{199}{342}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{55}{114}\right)\) |
| \(\chi_{3971}(846,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{329}{342}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) |
| \(\chi_{3971}(857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{59}{342}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) |
| \(\chi_{3971}(868,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{317}{342}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{259}{342}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{67}{114}\right)\) |
| \(\chi_{3971}(934,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{101}{342}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{5}{114}\right)\) |
| \(\chi_{3971}(945,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{197}{342}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{47}{114}\right)\) |
| \(\chi_{3971}(1022,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{305}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{49}{114}\right)\) |
| \(\chi_{3971}(1066,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{77}{114}\right)\) |
| \(\chi_{3971}(1077,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{25}{114}\right)\) |