from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([139]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,722))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(722\) | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{722}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{58}{171}\right)\) |
| \(\chi_{722}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{77}{171}\right)\) |
| \(\chi_{722}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{65}{171}\right)\) |
| \(\chi_{722}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{64}{171}\right)\) |
| \(\chi_{722}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{125}{342}\right)\) | \(e\left(\frac{44}{171}\right)\) |
| \(\chi_{722}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{342}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{287}{342}\right)\) | \(e\left(\frac{149}{342}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{151}{171}\right)\) |
| \(\chi_{722}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{342}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{155}{342}\right)\) | \(e\left(\frac{233}{342}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{103}{171}\right)\) |
| \(\chi_{722}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{43}{342}\right)\) | \(e\left(\frac{325}{342}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{149}{342}\right)\) | \(e\left(\frac{140}{171}\right)\) |
| \(\chi_{722}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{289}{342}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{47}{342}\right)\) | \(e\left(\frac{74}{171}\right)\) |
| \(\chi_{722}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{342}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{41}{342}\right)\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{325}{342}\right)\) | \(e\left(\frac{46}{171}\right)\) |
| \(\chi_{722}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{143}{171}\right)\) |
| \(\chi_{722}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{341}{342}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{259}{342}\right)\) | \(e\left(\frac{124}{171}\right)\) |
| \(\chi_{722}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{197}{342}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{148}{171}\right)\) |
| \(\chi_{722}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{259}{342}\right)\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{293}{342}\right)\) | \(e\left(\frac{32}{171}\right)\) |
| \(\chi_{722}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{342}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{263}{342}\right)\) | \(e\left(\frac{83}{171}\right)\) |
| \(\chi_{722}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{342}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{41}{171}\right)\) | \(e\left(\frac{235}{342}\right)\) | \(e\left(\frac{28}{171}\right)\) |
| \(\chi_{722}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{342}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{61}{342}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{89}{342}\right)\) | \(e\left(\frac{71}{171}\right)\) |
| \(\chi_{722}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{342}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{53}{342}\right)\) | \(e\left(\frac{329}{342}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{295}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) |
| \(\chi_{722}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{342}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{317}{342}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{319}{342}\right)\) | \(e\left(\frac{22}{171}\right)\) |
| \(\chi_{722}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) |
| \(\chi_{722}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{10}{171}\right)\) |
| \(\chi_{722}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{342}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{71}{342}\right)\) | \(e\left(\frac{170}{171}\right)\) |
| \(\chi_{722}(147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{70}{171}\right)\) |
| \(\chi_{722}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{125}{342}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{31}{342}\right)\) | \(e\left(\frac{67}{171}\right)\) |
| \(\chi_{722}(165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{342}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{7}{342}\right)\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{239}{342}\right)\) | \(e\left(\frac{158}{171}\right)\) |
| \(\chi_{722}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{342}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{235}{342}\right)\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{101}{171}\right)\) |
| \(\chi_{722}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{342}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{149}{342}\right)\) | \(e\left(\frac{299}{342}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{163}{171}\right)\) |
| \(\chi_{722}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{342}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{53}{342}\right)\) | \(e\left(\frac{98}{171}\right)\) |
| \(\chi_{722}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{342}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{43}{171}\right)\) |
| \(\chi_{722}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{342}\right)\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{89}{342}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) |
| \(\chi_{722}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{342}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{41}{342}\right)\) | \(e\left(\frac{50}{171}\right)\) |