from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([322,207,630]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,6815))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6815\) | |
| Conductor: | \(6815\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(644\) | 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: | $\Q(\zeta_{644})$ |
| Fixed field: | Number field defined by a degree 644 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6815}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{644}\right)\) | \(e\left(\frac{433}{644}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{569}{644}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{15}{322}\right)\) |
| \(\chi_{6815}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{291}{644}\right)\) | \(e\left(\frac{55}{644}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{229}{644}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{163}{644}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{225}{322}\right)\) |
| \(\chi_{6815}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{11}{644}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{561}{644}\right)\) | \(e\left(\frac{11}{322}\right)\) | \(e\left(\frac{419}{644}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{45}{322}\right)\) |
| \(\chi_{6815}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{339}{644}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{81}{644}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{177}{322}\right)\) |
| \(\chi_{6815}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{171}{644}\right)\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{349}{644}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{319}{322}\right)\) |
| \(\chi_{6815}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{559}{644}\right)\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{389}{644}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{395}{644}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{243}{322}\right)\) |
| \(\chi_{6815}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{267}{644}\right)\) | \(e\left(\frac{243}{644}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{157}{644}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{123}{644}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{233}{322}\right)\) |
| \(\chi_{6815}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{501}{644}\right)\) | \(e\left(\frac{181}{644}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{269}{322}\right)\) | \(e\left(\frac{215}{644}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{513}{644}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{155}{322}\right)\) |
| \(\chi_{6815}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{644}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{145}{644}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{127}{322}\right)\) |
| \(\chi_{6815}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{477}{644}\right)\) | \(e\left(\frac{369}{644}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{473}{644}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{163}{322}\right)\) |
| \(\chi_{6815}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{644}\right)\) | \(e\left(\frac{267}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{93}{644}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{159}{644}\right)\) | \(e\left(\frac{47}{92}\right)\) | \(e\left(\frac{97}{322}\right)\) |
| \(\chi_{6815}(264,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{644}\right)\) | \(e\left(\frac{393}{644}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{27}{322}\right)\) |
| \(\chi_{6815}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{463}{644}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{283}{322}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{101}{644}\right)\) | \(e\left(\frac{103}{322}\right)\) | \(e\left(\frac{235}{644}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{275}{322}\right)\) |
| \(\chi_{6815}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{644}\right)\) | \(e\left(\frac{445}{644}\right)\) | \(e\left(\frac{159}{322}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{155}{644}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{265}{644}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{269}{322}\right)\) |
| \(\chi_{6815}(334,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{551}{644}\right)\) | \(e\left(\frac{487}{644}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{31}{322}\right)\) |
| \(\chi_{6815}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{644}\right)\) | \(e\left(\frac{593}{644}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{165}{644}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{289}{322}\right)\) |
| \(\chi_{6815}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{289}{644}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{353}{644}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{187}{322}\right)\) |
| \(\chi_{6815}(374,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{463}{644}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{303}{322}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{429}{644}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{131}{644}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{167}{322}\right)\) |
| \(\chi_{6815}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{644}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{137}{322}\right)\) | \(e\left(\frac{471}{644}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{369}{644}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{55}{322}\right)\) |
| \(\chi_{6815}(414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{275}{644}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{501}{644}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{171}{644}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{159}{322}\right)\) |
| \(\chi_{6815}(449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{644}\right)\) | \(e\left(\frac{277}{644}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{603}{644}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{13}{644}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{255}{322}\right)\) |
| \(\chi_{6815}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{641}{644}\right)\) | \(e\left(\frac{265}{644}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{635}{644}\right)\) | \(e\left(\frac{265}{322}\right)\) | \(e\left(\frac{317}{644}\right)\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{1}{322}\right)\) |
| \(\chi_{6815}(514,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{601}{644}\right)\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{515}{644}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{465}{644}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{229}{322}\right)\) |
| \(\chi_{6815}(569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{579}{644}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{549}{644}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{627}{644}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{261}{322}\right)\) |
| \(\chi_{6815}(594,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{644}\right)\) | \(e\left(\frac{501}{644}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{65}{322}\right)\) | \(e\left(\frac{435}{644}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{349}{644}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{59}{322}\right)\) |
| \(\chi_{6815}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{473}{644}\right)\) | \(e\left(\frac{293}{644}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{131}{644}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{37}{644}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{57}{322}\right)\) |
| \(\chi_{6815}(624,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{495}{644}\right)\) | \(e\left(\frac{67}{644}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{197}{644}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{503}{644}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{157}{322}\right)\) |
| \(\chi_{6815}(649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{644}\right)\) | \(e\left(\frac{229}{644}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{87}{644}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{585}{644}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{205}{322}\right)\) |
| \(\chi_{6815}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{644}\right)\) | \(e\left(\frac{465}{644}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{531}{644}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{617}{644}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{263}{322}\right)\) |
| \(\chi_{6815}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{644}\right)\) | \(e\left(\frac{589}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{415}{644}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{481}{644}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{97}{322}\right)\) |
| \(\chi_{6815}(744,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{613}{644}\right)\) | \(e\left(\frac{377}{644}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{551}{644}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{485}{644}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{225}{322}\right)\) |