from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(546))
M = H._module
chi = DirichletCharacter(H, M([130,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,8281))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8281\) | |
| Conductor: | \(8281\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(546\) | 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: | $\Q(\zeta_{273})$ |
| Fixed field: | Number field defined by a degree 546 polynomial (not computed) |
First 28 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8281}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{182}\right)\) | \(e\left(\frac{226}{273}\right)\) | \(e\left(\frac{37}{91}\right)\) | \(e\left(\frac{11}{546}\right)\) | \(e\left(\frac{17}{546}\right)\) | \(e\left(\frac{111}{182}\right)\) | \(e\left(\frac{179}{273}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{461}{546}\right)\) | \(e\left(\frac{64}{273}\right)\) |
| \(\chi_{8281}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{182}\right)\) | \(e\left(\frac{94}{273}\right)\) | \(e\left(\frac{17}{91}\right)\) | \(e\left(\frac{251}{546}\right)\) | \(e\left(\frac{239}{546}\right)\) | \(e\left(\frac{51}{182}\right)\) | \(e\left(\frac{188}{273}\right)\) | \(e\left(\frac{151}{273}\right)\) | \(e\left(\frac{443}{546}\right)\) | \(e\left(\frac{145}{273}\right)\) |
| \(\chi_{8281}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{182}\right)\) | \(e\left(\frac{137}{273}\right)\) | \(e\left(\frac{18}{91}\right)\) | \(e\left(\frac{421}{546}\right)\) | \(e\left(\frac{55}{546}\right)\) | \(e\left(\frac{145}{182}\right)\) | \(e\left(\frac{1}{273}\right)\) | \(e\left(\frac{101}{273}\right)\) | \(e\left(\frac{271}{546}\right)\) | \(e\left(\frac{191}{273}\right)\) |
| \(\chi_{8281}(186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{182}\right)\) | \(e\left(\frac{235}{273}\right)\) | \(e\left(\frac{88}{91}\right)\) | \(e\left(\frac{491}{546}\right)\) | \(e\left(\frac{461}{546}\right)\) | \(e\left(\frac{173}{182}\right)\) | \(e\left(\frac{197}{273}\right)\) | \(e\left(\frac{241}{273}\right)\) | \(e\left(\frac{425}{546}\right)\) | \(e\left(\frac{226}{273}\right)\) |
| \(\chi_{8281}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{182}\right)\) | \(e\left(\frac{209}{273}\right)\) | \(e\left(\frac{62}{91}\right)\) | \(e\left(\frac{439}{546}\right)\) | \(e\left(\frac{331}{546}\right)\) | \(e\left(\frac{95}{182}\right)\) | \(e\left(\frac{145}{273}\right)\) | \(e\left(\frac{176}{273}\right)\) | \(e\left(\frac{529}{546}\right)\) | \(e\left(\frac{122}{273}\right)\) |
| \(\chi_{8281}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{182}\right)\) | \(e\left(\frac{103}{273}\right)\) | \(e\left(\frac{68}{91}\right)\) | \(e\left(\frac{185}{546}\right)\) | \(e\left(\frac{137}{546}\right)\) | \(e\left(\frac{113}{182}\right)\) | \(e\left(\frac{206}{273}\right)\) | \(e\left(\frac{58}{273}\right)\) | \(e\left(\frac{407}{546}\right)\) | \(e\left(\frac{34}{273}\right)\) |
| \(\chi_{8281}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{182}\right)\) | \(e\left(\frac{8}{273}\right)\) | \(e\left(\frac{15}{91}\right)\) | \(e\left(\frac{457}{546}\right)\) | \(e\left(\frac{61}{546}\right)\) | \(e\left(\frac{45}{182}\right)\) | \(e\left(\frac{16}{273}\right)\) | \(e\left(\frac{251}{273}\right)\) | \(e\left(\frac{241}{546}\right)\) | \(e\left(\frac{53}{273}\right)\) |
| \(\chi_{8281}(368,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{182}\right)\) | \(e\left(\frac{244}{273}\right)\) | \(e\left(\frac{48}{91}\right)\) | \(e\left(\frac{425}{546}\right)\) | \(e\left(\frac{359}{546}\right)\) | \(e\left(\frac{53}{182}\right)\) | \(e\left(\frac{215}{273}\right)\) | \(e\left(\frac{148}{273}\right)\) | \(e\left(\frac{389}{546}\right)\) | \(e\left(\frac{115}{273}\right)\) |
| \(\chi_{8281}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{80}{273}\right)\) | \(e\left(\frac{59}{91}\right)\) | \(e\left(\frac{475}{546}\right)\) | \(e\left(\frac{337}{546}\right)\) | \(e\left(\frac{177}{182}\right)\) | \(e\left(\frac{160}{273}\right)\) | \(e\left(\frac{53}{273}\right)\) | \(e\left(\frac{499}{546}\right)\) | \(e\left(\frac{257}{273}\right)\) |
| \(\chi_{8281}(478,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{182}\right)\) | \(e\left(\frac{152}{273}\right)\) | \(e\left(\frac{12}{91}\right)\) | \(e\left(\frac{493}{546}\right)\) | \(e\left(\frac{67}{546}\right)\) | \(e\left(\frac{127}{182}\right)\) | \(e\left(\frac{31}{273}\right)\) | \(e\left(\frac{128}{273}\right)\) | \(e\left(\frac{211}{546}\right)\) | \(e\left(\frac{188}{273}\right)\) |
| \(\chi_{8281}(550,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{182}\right)\) | \(e\left(\frac{253}{273}\right)\) | \(e\left(\frac{8}{91}\right)\) | \(e\left(\frac{359}{546}\right)\) | \(e\left(\frac{257}{546}\right)\) | \(e\left(\frac{115}{182}\right)\) | \(e\left(\frac{233}{273}\right)\) | \(e\left(\frac{55}{273}\right)\) | \(e\left(\frac{353}{546}\right)\) | \(e\left(\frac{4}{273}\right)\) |
| \(\chi_{8281}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{182}\right)\) | \(e\left(\frac{121}{273}\right)\) | \(e\left(\frac{79}{91}\right)\) | \(e\left(\frac{53}{546}\right)\) | \(e\left(\frac{479}{546}\right)\) | \(e\left(\frac{55}{182}\right)\) | \(e\left(\frac{242}{273}\right)\) | \(e\left(\frac{145}{273}\right)\) | \(e\left(\frac{335}{546}\right)\) | \(e\left(\frac{85}{273}\right)\) |
| \(\chi_{8281}(660,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{182}\right)\) | \(e\left(\frac{23}{273}\right)\) | \(e\left(\frac{9}{91}\right)\) | \(e\left(\frac{529}{546}\right)\) | \(e\left(\frac{73}{546}\right)\) | \(e\left(\frac{27}{182}\right)\) | \(e\left(\frac{46}{273}\right)\) | \(e\left(\frac{5}{273}\right)\) | \(e\left(\frac{181}{546}\right)\) | \(e\left(\frac{50}{273}\right)\) |
| \(\chi_{8281}(732,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{262}{273}\right)\) | \(e\left(\frac{59}{91}\right)\) | \(e\left(\frac{293}{546}\right)\) | \(e\left(\frac{155}{546}\right)\) | \(e\left(\frac{177}{182}\right)\) | \(e\left(\frac{251}{273}\right)\) | \(e\left(\frac{235}{273}\right)\) | \(e\left(\frac{317}{546}\right)\) | \(e\left(\frac{166}{273}\right)\) |
| \(\chi_{8281}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{182}\right)\) | \(e\left(\frac{95}{273}\right)\) | \(e\left(\frac{53}{91}\right)\) | \(e\left(\frac{1}{546}\right)\) | \(e\left(\frac{349}{546}\right)\) | \(e\left(\frac{159}{182}\right)\) | \(e\left(\frac{190}{273}\right)\) | \(e\left(\frac{80}{273}\right)\) | \(e\left(\frac{439}{546}\right)\) | \(e\left(\frac{254}{273}\right)\) |
| \(\chi_{8281}(842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{182}\right)\) | \(e\left(\frac{167}{273}\right)\) | \(e\left(\frac{6}{91}\right)\) | \(e\left(\frac{19}{546}\right)\) | \(e\left(\frac{79}{546}\right)\) | \(e\left(\frac{109}{182}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{155}{273}\right)\) | \(e\left(\frac{151}{546}\right)\) | \(e\left(\frac{185}{273}\right)\) |
| \(\chi_{8281}(914,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{182}\right)\) | \(e\left(\frac{271}{273}\right)\) | \(e\left(\frac{19}{91}\right)\) | \(e\left(\frac{227}{546}\right)\) | \(e\left(\frac{53}{546}\right)\) | \(e\left(\frac{57}{182}\right)\) | \(e\left(\frac{269}{273}\right)\) | \(e\left(\frac{142}{273}\right)\) | \(e\left(\frac{281}{546}\right)\) | \(e\left(\frac{55}{273}\right)\) |
| \(\chi_{8281}(933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{239}{273}\right)\) | \(e\left(\frac{50}{91}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{355}{546}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{205}{273}\right)\) | \(e\left(\frac{230}{273}\right)\) | \(e\left(\frac{409}{546}\right)\) | \(e\left(\frac{116}{273}\right)\) |
| \(\chi_{8281}(1005,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{182}\right)\) | \(e\left(\frac{139}{273}\right)\) | \(e\left(\frac{90}{91}\right)\) | \(e\left(\frac{467}{546}\right)\) | \(e\left(\frac{275}{546}\right)\) | \(e\left(\frac{179}{182}\right)\) | \(e\left(\frac{5}{273}\right)\) | \(e\left(\frac{232}{273}\right)\) | \(e\left(\frac{263}{546}\right)\) | \(e\left(\frac{136}{273}\right)\) |
| \(\chi_{8281}(1024,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{182}\right)\) | \(e\left(\frac{38}{273}\right)\) | \(e\left(\frac{3}{91}\right)\) | \(e\left(\frac{55}{546}\right)\) | \(e\left(\frac{85}{546}\right)\) | \(e\left(\frac{9}{182}\right)\) | \(e\left(\frac{76}{273}\right)\) | \(e\left(\frac{32}{273}\right)\) | \(e\left(\frac{121}{546}\right)\) | \(e\left(\frac{47}{273}\right)\) |
| \(\chi_{8281}(1115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{182}\right)\) | \(e\left(\frac{110}{273}\right)\) | \(e\left(\frac{47}{91}\right)\) | \(e\left(\frac{73}{546}\right)\) | \(e\left(\frac{361}{546}\right)\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{220}{273}\right)\) | \(e\left(\frac{107}{273}\right)\) | \(e\left(\frac{379}{546}\right)\) | \(e\left(\frac{251}{273}\right)\) |
| \(\chi_{8281}(1187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{148}{273}\right)\) | \(e\left(\frac{50}{91}\right)\) | \(e\left(\frac{401}{546}\right)\) | \(e\left(\frac{173}{546}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{23}{273}\right)\) | \(e\left(\frac{139}{273}\right)\) | \(e\left(\frac{227}{546}\right)\) | \(e\left(\frac{25}{273}\right)\) |
| \(\chi_{8281}(1278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{182}\right)\) | \(e\left(\frac{16}{273}\right)\) | \(e\left(\frac{30}{91}\right)\) | \(e\left(\frac{95}{546}\right)\) | \(e\left(\frac{395}{546}\right)\) | \(e\left(\frac{181}{182}\right)\) | \(e\left(\frac{32}{273}\right)\) | \(e\left(\frac{229}{273}\right)\) | \(e\left(\frac{209}{546}\right)\) | \(e\left(\frac{106}{273}\right)\) |
| \(\chi_{8281}(1297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{182}\right)\) | \(e\left(\frac{254}{273}\right)\) | \(e\left(\frac{44}{91}\right)\) | \(e\left(\frac{109}{546}\right)\) | \(e\left(\frac{367}{546}\right)\) | \(e\left(\frac{41}{182}\right)\) | \(e\left(\frac{235}{273}\right)\) | \(e\left(\frac{257}{273}\right)\) | \(e\left(\frac{349}{546}\right)\) | \(e\left(\frac{113}{273}\right)\) |
| \(\chi_{8281}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{182}\right)\) | \(e\left(\frac{157}{273}\right)\) | \(e\left(\frac{10}{91}\right)\) | \(e\left(\frac{335}{546}\right)\) | \(e\left(\frac{71}{546}\right)\) | \(e\left(\frac{121}{182}\right)\) | \(e\left(\frac{41}{273}\right)\) | \(e\left(\frac{46}{273}\right)\) | \(e\left(\frac{191}{546}\right)\) | \(e\left(\frac{187}{273}\right)\) |
| \(\chi_{8281}(1388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{182}\right)\) | \(e\left(\frac{53}{273}\right)\) | \(e\left(\frac{88}{91}\right)\) | \(e\left(\frac{127}{546}\right)\) | \(e\left(\frac{97}{546}\right)\) | \(e\left(\frac{173}{182}\right)\) | \(e\left(\frac{106}{273}\right)\) | \(e\left(\frac{59}{273}\right)\) | \(e\left(\frac{61}{546}\right)\) | \(e\left(\frac{44}{273}\right)\) |
| \(\chi_{8281}(1460,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{182}\right)\) | \(e\left(\frac{25}{273}\right)\) | \(e\left(\frac{81}{91}\right)\) | \(e\left(\frac{29}{546}\right)\) | \(e\left(\frac{293}{546}\right)\) | \(e\left(\frac{61}{182}\right)\) | \(e\left(\frac{50}{273}\right)\) | \(e\left(\frac{136}{273}\right)\) | \(e\left(\frac{173}{546}\right)\) | \(e\left(\frac{268}{273}\right)\) |
| \(\chi_{8281}(1479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{182}\right)\) | \(e\left(\frac{125}{273}\right)\) | \(e\left(\frac{41}{91}\right)\) | \(e\left(\frac{145}{546}\right)\) | \(e\left(\frac{373}{546}\right)\) | \(e\left(\frac{123}{182}\right)\) | \(e\left(\frac{250}{273}\right)\) | \(e\left(\frac{134}{273}\right)\) | \(e\left(\frac{319}{546}\right)\) | \(e\left(\frac{248}{273}\right)\) |