from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([0,391,84]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,6815))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6815\) | |
| Conductor: | \(1363\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(644\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1363.w | 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: | $\Q(\zeta_{644})$ |
| Fixed field: | Number field defined by a degree 644 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6815}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{415}{644}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{557}{644}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{59}{644}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{117}{322}\right)\) |
| \(\chi_{6815}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{644}\right)\) | \(e\left(\frac{45}{644}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{363}{644}\right)\) | \(e\left(\frac{45}{322}\right)\) | \(e\left(\frac{309}{644}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{67}{322}\right)\) |
| \(\chi_{6815}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{644}\right)\) | \(e\left(\frac{407}{644}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{47}{644}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{55}{322}\right)\) |
| \(\chi_{6815}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{635}{644}\right)\) | \(e\left(\frac{151}{644}\right)\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{617}{644}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{3}{322}\right)\) |
| \(\chi_{6815}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{644}\right)\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{453}{644}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{359}{644}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{57}{322}\right)\) |
| \(\chi_{6815}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{644}\right)\) | \(e\left(\frac{153}{644}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{99}{322}\right)\) |
| \(\chi_{6815}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{53}{644}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{127}{644}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{129}{322}\right)\) |
| \(\chi_{6815}(166,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{644}\right)\) | \(e\left(\frac{583}{644}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{109}{644}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{311}{644}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{131}{322}\right)\) |
| \(\chi_{6815}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{644}\right)\) | \(e\left(\frac{71}{644}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{401}{644}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{27}{322}\right)\) |
| \(\chi_{6815}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{319}{644}\right)\) | \(e\left(\frac{587}{644}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{313}{644}\right)\) | \(e\left(\frac{265}{322}\right)\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{1}{322}\right)\) |
| \(\chi_{6815}(271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{644}\right)\) | \(e\left(\frac{97}{644}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{117}{161}\right)\) | \(e\left(\frac{439}{644}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{65}{644}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{309}{322}\right)\) |
| \(\chi_{6815}(316,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{633}{644}\right)\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{611}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{89}{644}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{111}{322}\right)\) |
| \(\chi_{6815}(346,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{513}{644}\right)\) | \(e\left(\frac{409}{644}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{251}{644}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{533}{644}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{151}{322}\right)\) |
| \(\chi_{6815}(356,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{644}\right)\) | \(e\left(\frac{401}{644}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{4}{161}\right)\) | \(e\left(\frac{487}{644}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{521}{644}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{89}{322}\right)\) |
| \(\chi_{6815}(366,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{533}{644}\right)\) | \(e\left(\frac{145}{644}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{150}{161}\right)\) | \(e\left(\frac{311}{644}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{137}{644}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{37}{322}\right)\) |
| \(\chi_{6815}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{569}{644}\right)\) | \(e\left(\frac{185}{644}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{97}{161}\right)\) | \(e\left(\frac{419}{644}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{197}{644}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{25}{322}\right)\) |
| \(\chi_{6815}(541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{644}\right)\) | \(e\left(\frac{503}{644}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{191}{644}\right)\) | \(e\left(\frac{31}{92}\right)\) | \(e\left(\frac{155}{322}\right)\) |
| \(\chi_{6815}(566,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{644}\right)\) | \(e\left(\frac{417}{644}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{545}{644}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{213}{322}\right)\) |
| \(\chi_{6815}(591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{355}{644}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{73}{644}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{291}{644}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{135}{322}\right)\) |
| \(\chi_{6815}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{644}\right)\) | \(e\left(\frac{191}{644}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{81}{644}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{367}{644}\right)\) | \(e\left(\frac{35}{92}\right)\) | \(e\left(\frac{313}{322}\right)\) |
| \(\chi_{6815}(606,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{644}\right)\) | \(e\left(\frac{533}{644}\right)\) | \(e\left(\frac{45}{322}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{135}{644}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{397}{644}\right)\) | \(e\left(\frac{89}{92}\right)\) | \(e\left(\frac{307}{322}\right)\) |
| \(\chi_{6815}(636,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{644}\right)\) | \(e\left(\frac{353}{644}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{449}{644}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{39}{322}\right)\) |
| \(\chi_{6815}(686,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{599}{644}\right)\) | \(e\left(\frac{111}{644}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{247}{644}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{15}{322}\right)\) |
| \(\chi_{6815}(711,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{221}{644}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{323}{644}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{143}{322}\right)\) |
| \(\chi_{6815}(756,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{535}{644}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{237}{644}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{81}{322}\right)\) |
| \(\chi_{6815}(786,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{644}\right)\) | \(e\left(\frac{435}{644}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{289}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{411}{644}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{111}{322}\right)\) |
| \(\chi_{6815}(801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{199}{644}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{361}{644}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{121}{322}\right)\) |
| \(\chi_{6815}(826,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{607}{644}\right)\) | \(e\left(\frac{263}{644}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{533}{644}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{475}{644}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{227}{322}\right)\) |
| \(\chi_{6815}(831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{644}\right)\) | \(e\left(\frac{475}{644}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{397}{644}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{471}{644}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{99}{322}\right)\) |
| \(\chi_{6815}(896,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{644}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{5}{644}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{321}{322}\right)\) |
| \(\chi_{6815}(901,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{644}\right)\) | \(e\left(\frac{423}{644}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{265}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{71}{644}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{179}{322}\right)\) |