from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8009, base_ring=CyclotomicField(1144))
M = H._module
chi = DirichletCharacter(H, M([665]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,8009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8009\) | |
| Conductor: | \(8009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1144\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1144})$ |
| Fixed field: | Number field defined by a degree 1144 polynomial (not computed) |
First 31 of 480 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8009}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{572}\right)\) | \(e\left(\frac{665}{1144}\right)\) | \(e\left(\frac{61}{286}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{215}{1144}\right)\) | \(e\left(\frac{56}{143}\right)\) | \(e\left(\frac{469}{572}\right)\) | \(e\left(\frac{93}{572}\right)\) | \(e\left(\frac{271}{286}\right)\) | \(e\left(\frac{263}{572}\right)\) |
| \(\chi_{8009}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{572}\right)\) | \(e\left(\frac{463}{1144}\right)\) | \(e\left(\frac{153}{286}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{769}{1144}\right)\) | \(e\left(\frac{42}{143}\right)\) | \(e\left(\frac{459}{572}\right)\) | \(e\left(\frac{463}{572}\right)\) | \(e\left(\frac{239}{286}\right)\) | \(e\left(\frac{233}{572}\right)\) |
| \(\chi_{8009}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{447}{572}\right)\) | \(e\left(\frac{1005}{1144}\right)\) | \(e\left(\frac{161}{286}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{755}{1144}\right)\) | \(e\left(\frac{47}{143}\right)\) | \(e\left(\frac{197}{572}\right)\) | \(e\left(\frac{433}{572}\right)\) | \(e\left(\frac{87}{286}\right)\) | \(e\left(\frac{19}{572}\right)\) |
| \(\chi_{8009}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{572}\right)\) | \(e\left(\frac{683}{1144}\right)\) | \(e\left(\frac{285}{286}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{109}{1144}\right)\) | \(e\left(\frac{53}{143}\right)\) | \(e\left(\frac{283}{572}\right)\) | \(e\left(\frac{111}{572}\right)\) | \(e\left(\frac{19}{286}\right)\) | \(e\left(\frac{277}{572}\right)\) |
| \(\chi_{8009}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{572}\right)\) | \(e\left(\frac{95}{1144}\right)\) | \(e\left(\frac{213}{286}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{521}{1144}\right)\) | \(e\left(\frac{8}{143}\right)\) | \(e\left(\frac{67}{572}\right)\) | \(e\left(\frac{95}{572}\right)\) | \(e\left(\frac{243}{286}\right)\) | \(e\left(\frac{201}{572}\right)\) |
| \(\chi_{8009}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{467}{572}\right)\) | \(e\left(\frac{1073}{1144}\right)\) | \(e\left(\frac{181}{286}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{863}{1144}\right)\) | \(e\left(\frac{131}{143}\right)\) | \(e\left(\frac{257}{572}\right)\) | \(e\left(\frac{501}{572}\right)\) | \(e\left(\frac{279}{286}\right)\) | \(e\left(\frac{199}{572}\right)\) |
| \(\chi_{8009}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{371}{572}\right)\) | \(e\left(\frac{289}{1144}\right)\) | \(e\left(\frac{85}{286}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{1031}{1144}\right)\) | \(e\left(\frac{71}{143}\right)\) | \(e\left(\frac{541}{572}\right)\) | \(e\left(\frac{289}{572}\right)\) | \(e\left(\frac{101}{286}\right)\) | \(e\left(\frac{479}{572}\right)\) |
| \(\chi_{8009}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{572}\right)\) | \(e\left(\frac{883}{1144}\right)\) | \(e\left(\frac{41}{286}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{965}{1144}\right)\) | \(e\left(\frac{115}{143}\right)\) | \(e\left(\frac{123}{572}\right)\) | \(e\left(\frac{311}{572}\right)\) | \(e\left(\frac{79}{286}\right)\) | \(e\left(\frac{369}{572}\right)\) |
| \(\chi_{8009}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{572}\right)\) | \(e\left(\frac{1129}{1144}\right)\) | \(e\left(\frac{147}{286}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{279}{1144}\right)\) | \(e\left(\frac{74}{143}\right)\) | \(e\left(\frac{441}{572}\right)\) | \(e\left(\frac{557}{572}\right)\) | \(e\left(\frac{67}{286}\right)\) | \(e\left(\frac{179}{572}\right)\) |
| \(\chi_{8009}(132,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{572}\right)\) | \(e\left(\frac{525}{1144}\right)\) | \(e\left(\frac{3}{286}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{531}{1144}\right)\) | \(e\left(\frac{127}{143}\right)\) | \(e\left(\frac{9}{572}\right)\) | \(e\left(\frac{525}{572}\right)\) | \(e\left(\frac{229}{286}\right)\) | \(e\left(\frac{27}{572}\right)\) |
| \(\chi_{8009}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{572}\right)\) | \(e\left(\frac{43}{1144}\right)\) | \(e\left(\frac{265}{286}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{573}{1144}\right)\) | \(e\left(\frac{112}{143}\right)\) | \(e\left(\frac{223}{572}\right)\) | \(e\left(\frac{43}{572}\right)\) | \(e\left(\frac{113}{286}\right)\) | \(e\left(\frac{97}{572}\right)\) |
| \(\chi_{8009}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{415}{572}\right)\) | \(e\left(\frac{553}{1144}\right)\) | \(e\left(\frac{129}{286}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{239}{1144}\right)\) | \(e\left(\frac{27}{143}\right)\) | \(e\left(\frac{101}{572}\right)\) | \(e\left(\frac{553}{572}\right)\) | \(e\left(\frac{123}{286}\right)\) | \(e\left(\frac{303}{572}\right)\) |
| \(\chi_{8009}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{437}{572}\right)\) | \(e\left(\frac{399}{1144}\right)\) | \(e\left(\frac{151}{286}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{129}{1144}\right)\) | \(e\left(\frac{5}{143}\right)\) | \(e\left(\frac{167}{572}\right)\) | \(e\left(\frac{399}{572}\right)\) | \(e\left(\frac{277}{286}\right)\) | \(e\left(\frac{501}{572}\right)\) |
| \(\chi_{8009}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{572}\right)\) | \(e\left(\frac{677}{1144}\right)\) | \(e\left(\frac{115}{286}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{907}{1144}\right)\) | \(e\left(\frac{54}{143}\right)\) | \(e\left(\frac{345}{572}\right)\) | \(e\left(\frac{105}{572}\right)\) | \(e\left(\frac{103}{286}\right)\) | \(e\left(\frac{463}{572}\right)\) |
| \(\chi_{8009}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{501}{572}\right)\) | \(e\left(\frac{731}{1144}\right)\) | \(e\left(\frac{215}{286}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{589}{1144}\right)\) | \(e\left(\frac{45}{143}\right)\) | \(e\left(\frac{359}{572}\right)\) | \(e\left(\frac{159}{572}\right)\) | \(e\left(\frac{205}{286}\right)\) | \(e\left(\frac{505}{572}\right)\) |
| \(\chi_{8009}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{572}\right)\) | \(e\left(\frac{595}{1144}\right)\) | \(e\left(\frac{175}{286}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{373}{1144}\right)\) | \(e\left(\frac{20}{143}\right)\) | \(e\left(\frac{239}{572}\right)\) | \(e\left(\frac{23}{572}\right)\) | \(e\left(\frac{107}{286}\right)\) | \(e\left(\frac{145}{572}\right)\) |
| \(\chi_{8009}(258,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{572}\right)\) | \(e\left(\frac{257}{1144}\right)\) | \(e\left(\frac{227}{286}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{711}{1144}\right)\) | \(e\left(\frac{124}{143}\right)\) | \(e\left(\frac{109}{572}\right)\) | \(e\left(\frac{257}{572}\right)\) | \(e\left(\frac{263}{286}\right)\) | \(e\left(\frac{327}{572}\right)\) |
| \(\chi_{8009}(273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{493}{572}\right)\) | \(e\left(\frac{475}{1144}\right)\) | \(e\left(\frac{207}{286}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{317}{1144}\right)\) | \(e\left(\frac{40}{143}\right)\) | \(e\left(\frac{335}{572}\right)\) | \(e\left(\frac{475}{572}\right)\) | \(e\left(\frac{71}{286}\right)\) | \(e\left(\frac{433}{572}\right)\) |
| \(\chi_{8009}(287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{267}{572}\right)\) | \(e\left(\frac{393}{1144}\right)\) | \(e\left(\frac{267}{286}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{927}{1144}\right)\) | \(e\left(\frac{6}{143}\right)\) | \(e\left(\frac{229}{572}\right)\) | \(e\left(\frac{393}{572}\right)\) | \(e\left(\frac{75}{286}\right)\) | \(e\left(\frac{115}{572}\right)\) |
| \(\chi_{8009}(312,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{572}\right)\) | \(e\left(\frac{1017}{1144}\right)\) | \(e\left(\frac{215}{286}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{303}{1144}\right)\) | \(e\left(\frac{45}{143}\right)\) | \(e\left(\frac{73}{572}\right)\) | \(e\left(\frac{445}{572}\right)\) | \(e\left(\frac{205}{286}\right)\) | \(e\left(\frac{219}{572}\right)\) |
| \(\chi_{8009}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{572}\right)\) | \(e\left(\frac{1011}{1144}\right)\) | \(e\left(\frac{45}{286}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{1101}{1144}\right)\) | \(e\left(\frac{46}{143}\right)\) | \(e\left(\frac{135}{572}\right)\) | \(e\left(\frac{439}{572}\right)\) | \(e\left(\frac{3}{286}\right)\) | \(e\left(\frac{405}{572}\right)\) |
| \(\chi_{8009}(418,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{413}{572}\right)\) | \(e\left(\frac{203}{1144}\right)\) | \(e\left(\frac{127}{286}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{1029}{1144}\right)\) | \(e\left(\frac{133}{143}\right)\) | \(e\left(\frac{95}{572}\right)\) | \(e\left(\frac{203}{572}\right)\) | \(e\left(\frac{161}{286}\right)\) | \(e\left(\frac{285}{572}\right)\) |
| \(\chi_{8009}(423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{572}\right)\) | \(e\left(\frac{357}{1144}\right)\) | \(e\left(\frac{105}{286}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{1139}{1144}\right)\) | \(e\left(\frac{12}{143}\right)\) | \(e\left(\frac{29}{572}\right)\) | \(e\left(\frac{357}{572}\right)\) | \(e\left(\frac{7}{286}\right)\) | \(e\left(\frac{87}{572}\right)\) |
| \(\chi_{8009}(426,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{572}\right)\) | \(e\left(\frac{643}{1144}\right)\) | \(e\left(\frac{105}{286}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{853}{1144}\right)\) | \(e\left(\frac{12}{143}\right)\) | \(e\left(\frac{315}{572}\right)\) | \(e\left(\frac{71}{572}\right)\) | \(e\left(\frac{7}{286}\right)\) | \(e\left(\frac{373}{572}\right)\) |
| \(\chi_{8009}(449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{375}{572}\right)\) | \(e\left(\frac{417}{1144}\right)\) | \(e\left(\frac{89}{286}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{23}{1144}\right)\) | \(e\left(\frac{2}{143}\right)\) | \(e\left(\frac{553}{572}\right)\) | \(e\left(\frac{417}{572}\right)\) | \(e\left(\frac{25}{286}\right)\) | \(e\left(\frac{515}{572}\right)\) |
| \(\chi_{8009}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{572}\right)\) | \(e\left(\frac{305}{1144}\right)\) | \(e\left(\frac{157}{286}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{47}{1144}\right)\) | \(e\left(\frac{116}{143}\right)\) | \(e\left(\frac{185}{572}\right)\) | \(e\left(\frac{305}{572}\right)\) | \(e\left(\frac{163}{286}\right)\) | \(e\left(\frac{555}{572}\right)\) |
| \(\chi_{8009}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{463}{572}\right)\) | \(e\left(\frac{373}{1144}\right)\) | \(e\left(\frac{177}{286}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{155}{1144}\right)\) | \(e\left(\frac{57}{143}\right)\) | \(e\left(\frac{245}{572}\right)\) | \(e\left(\frac{373}{572}\right)\) | \(e\left(\frac{69}{286}\right)\) | \(e\left(\frac{163}{572}\right)\) |
| \(\chi_{8009}(486,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{572}\right)\) | \(e\left(\frac{717}{1144}\right)\) | \(e\left(\frac{9}{286}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{163}{1144}\right)\) | \(e\left(\frac{95}{143}\right)\) | \(e\left(\frac{313}{572}\right)\) | \(e\left(\frac{145}{572}\right)\) | \(e\left(\frac{115}{286}\right)\) | \(e\left(\frac{367}{572}\right)\) |
| \(\chi_{8009}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{572}\right)\) | \(e\left(\frac{1009}{1144}\right)\) | \(e\left(\frac{179}{286}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{223}{1144}\right)\) | \(e\left(\frac{94}{143}\right)\) | \(e\left(\frac{537}{572}\right)\) | \(e\left(\frac{437}{572}\right)\) | \(e\left(\frac{31}{286}\right)\) | \(e\left(\frac{467}{572}\right)\) |
| \(\chi_{8009}(489,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{572}\right)\) | \(e\left(\frac{599}{1144}\right)\) | \(e\left(\frac{193}{286}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{985}{1144}\right)\) | \(e\left(\frac{67}{143}\right)\) | \(e\left(\frac{7}{572}\right)\) | \(e\left(\frac{27}{572}\right)\) | \(e\left(\frac{51}{286}\right)\) | \(e\left(\frac{21}{572}\right)\) |
| \(\chi_{8009}(498,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{572}\right)\) | \(e\left(\frac{1069}{1144}\right)\) | \(e\left(\frac{163}{286}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{251}{1144}\right)\) | \(e\left(\frac{84}{143}\right)\) | \(e\left(\frac{489}{572}\right)\) | \(e\left(\frac{497}{572}\right)\) | \(e\left(\frac{49}{286}\right)\) | \(e\left(\frac{323}{572}\right)\) |