from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8664, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([171,171,0,52]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,8664))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8664\) | |
| Conductor: | \(2888\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(342\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2888.bs | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8664}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{29}{342}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{14}{171}\right)\) |
| \(\chi_{8664}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{342}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{163}{342}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{109}{171}\right)\) |
| \(\chi_{8664}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{275}{342}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{62}{171}\right)\) |
| \(\chi_{8664}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{59}{342}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{146}{171}\right)\) |
| \(\chi_{8664}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{163}{342}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{157}{171}\right)\) |
| \(\chi_{8664}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{52}{171}\right)\) |
| \(\chi_{8664}(499,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{342}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{215}{342}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{203}{342}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{77}{171}\right)\) |
| \(\chi_{8664}(643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{125}{342}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{257}{342}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{71}{171}\right)\) |
| \(\chi_{8664}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{74}{171}\right)\) |
| \(\chi_{8664}(859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{229}{342}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{241}{342}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{22}{171}\right)\) |
| \(\chi_{8664}(883,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{342}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{295}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{106}{171}\right)\) |
| \(\chi_{8664}(955,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{140}{171}\right)\) |
| \(\chi_{8664}(1051,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{305}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{217}{342}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{100}{171}\right)\) |
| \(\chi_{8664}(1099,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{71}{342}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{80}{171}\right)\) |
| \(\chi_{8664}(1195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{83}{342}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{53}{342}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{2}{171}\right)\) |
| \(\chi_{8664}(1315,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{342}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{58}{171}\right)\) |
| \(\chi_{8664}(1339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{342}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{160}{171}\right)\) |
| \(\chi_{8664}(1411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{287}{342}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{131}{342}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{32}{171}\right)\) |
| \(\chi_{8664}(1507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{342}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{79}{342}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{10}{171}\right)\) |
| \(\chi_{8664}(1555,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{342}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{89}{171}\right)\) |
| \(\chi_{8664}(1651,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{342}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{131}{342}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{197}{342}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{101}{171}\right)\) |
| \(\chi_{8664}(1771,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{127}{342}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{139}{342}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{97}{342}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{94}{171}\right)\) |
| \(\chi_{8664}(1795,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{342}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{49}{342}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{199}{342}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{43}{171}\right)\) |
| \(\chi_{8664}(1963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{91}{171}\right)\) |
| \(\chi_{8664}(2011,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{305}{342}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{203}{342}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{98}{171}\right)\) |
| \(\chi_{8664}(2107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{342}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{41}{342}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{263}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{341}{342}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{29}{171}\right)\) |
| \(\chi_{8664}(2227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{130}{171}\right)\) |
| \(\chi_{8664}(2251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{31}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{7}{342}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{91}{342}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{97}{171}\right)\) |
| \(\chi_{8664}(2323,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{59}{342}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{83}{342}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{158}{171}\right)\) |
| \(\chi_{8664}(2419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{127}{342}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{283}{342}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{1}{171}\right)\) |
| \(\chi_{8664}(2467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{281}{342}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{185}{342}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{107}{171}\right)\) |