from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1319, base_ring=CyclotomicField(1318))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1319))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1319\) | |
| Conductor: | \(1319\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1318\) | 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: | $\Q(\zeta_{659})$ |
| Fixed field: | Number field defined by a degree 1318 polynomial (not computed) |
First 31 of 658 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1319}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{659}\right)\) | \(e\left(\frac{346}{659}\right)\) | \(e\left(\frac{18}{659}\right)\) | \(e\left(\frac{604}{659}\right)\) | \(e\left(\frac{355}{659}\right)\) | \(e\left(\frac{149}{659}\right)\) | \(e\left(\frac{27}{659}\right)\) | \(e\left(\frac{33}{659}\right)\) | \(e\left(\frac{613}{659}\right)\) | \(e\left(\frac{341}{659}\right)\) |
| \(\chi_{1319}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{237}{659}\right)\) | \(e\left(\frac{105}{659}\right)\) | \(e\left(\frac{474}{659}\right)\) | \(e\left(\frac{309}{659}\right)\) | \(e\left(\frac{342}{659}\right)\) | \(e\left(\frac{409}{659}\right)\) | \(e\left(\frac{52}{659}\right)\) | \(e\left(\frac{210}{659}\right)\) | \(e\left(\frac{546}{659}\right)\) | \(e\left(\frac{193}{659}\right)\) |
| \(\chi_{1319}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{518}{659}\right)\) | \(e\left(\frac{71}{659}\right)\) | \(e\left(\frac{377}{659}\right)\) | \(e\left(\frac{642}{659}\right)\) | \(e\left(\frac{589}{659}\right)\) | \(e\left(\frac{82}{659}\right)\) | \(e\left(\frac{236}{659}\right)\) | \(e\left(\frac{142}{659}\right)\) | \(e\left(\frac{501}{659}\right)\) | \(e\left(\frac{369}{659}\right)\) |
| \(\chi_{1319}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{659}\right)\) | \(e\left(\frac{643}{659}\right)\) | \(e\left(\frac{342}{659}\right)\) | \(e\left(\frac{273}{659}\right)\) | \(e\left(\frac{155}{659}\right)\) | \(e\left(\frac{195}{659}\right)\) | \(e\left(\frac{513}{659}\right)\) | \(e\left(\frac{627}{659}\right)\) | \(e\left(\frac{444}{659}\right)\) | \(e\left(\frac{548}{659}\right)\) |
| \(\chi_{1319}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{604}{659}\right)\) | \(e\left(\frac{9}{659}\right)\) | \(e\left(\frac{549}{659}\right)\) | \(e\left(\frac{629}{659}\right)\) | \(e\left(\frac{613}{659}\right)\) | \(e\left(\frac{261}{659}\right)\) | \(e\left(\frac{494}{659}\right)\) | \(e\left(\frac{18}{659}\right)\) | \(e\left(\frac{574}{659}\right)\) | \(e\left(\frac{186}{659}\right)\) |
| \(\chi_{1319}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{399}{659}\right)\) | \(e\left(\frac{402}{659}\right)\) | \(e\left(\frac{139}{659}\right)\) | \(e\left(\frac{637}{659}\right)\) | \(e\left(\frac{142}{659}\right)\) | \(e\left(\frac{455}{659}\right)\) | \(e\left(\frac{538}{659}\right)\) | \(e\left(\frac{145}{659}\right)\) | \(e\left(\frac{377}{659}\right)\) | \(e\left(\frac{400}{659}\right)\) |
| \(\chi_{1319}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{659}\right)\) | \(e\left(\frac{640}{659}\right)\) | \(e\left(\frac{159}{659}\right)\) | \(e\left(\frac{283}{659}\right)\) | \(e\left(\frac{390}{659}\right)\) | \(e\left(\frac{108}{659}\right)\) | \(e\left(\frac{568}{659}\right)\) | \(e\left(\frac{621}{659}\right)\) | \(e\left(\frac{33}{659}\right)\) | \(e\left(\frac{486}{659}\right)\) |
| \(\chi_{1319}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{306}{659}\right)\) | \(e\left(\frac{561}{659}\right)\) | \(e\left(\frac{612}{659}\right)\) | \(e\left(\frac{107}{659}\right)\) | \(e\left(\frac{208}{659}\right)\) | \(e\left(\frac{453}{659}\right)\) | \(e\left(\frac{259}{659}\right)\) | \(e\left(\frac{463}{659}\right)\) | \(e\left(\frac{413}{659}\right)\) | \(e\left(\frac{391}{659}\right)\) |
| \(\chi_{1319}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{554}{659}\right)\) | \(e\left(\frac{137}{659}\right)\) | \(e\left(\frac{449}{659}\right)\) | \(e\left(\frac{422}{659}\right)\) | \(e\left(\frac{32}{659}\right)\) | \(e\left(\frac{19}{659}\right)\) | \(e\left(\frac{344}{659}\right)\) | \(e\left(\frac{274}{659}\right)\) | \(e\left(\frac{317}{659}\right)\) | \(e\left(\frac{415}{659}\right)\) |
| \(\chi_{1319}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{659}\right)\) | \(e\left(\frac{368}{659}\right)\) | \(e\left(\frac{42}{659}\right)\) | \(e\left(\frac{311}{659}\right)\) | \(e\left(\frac{389}{659}\right)\) | \(e\left(\frac{128}{659}\right)\) | \(e\left(\frac{63}{659}\right)\) | \(e\left(\frac{77}{659}\right)\) | \(e\left(\frac{332}{659}\right)\) | \(e\left(\frac{576}{659}\right)\) |
| \(\chi_{1319}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{586}{659}\right)\) | \(e\left(\frac{635}{659}\right)\) | \(e\left(\frac{513}{659}\right)\) | \(e\left(\frac{80}{659}\right)\) | \(e\left(\frac{562}{659}\right)\) | \(e\left(\frac{622}{659}\right)\) | \(e\left(\frac{440}{659}\right)\) | \(e\left(\frac{611}{659}\right)\) | \(e\left(\frac{7}{659}\right)\) | \(e\left(\frac{163}{659}\right)\) |
| \(\chi_{1319}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{534}{659}\right)\) | \(e\left(\frac{320}{659}\right)\) | \(e\left(\frac{409}{659}\right)\) | \(e\left(\frac{471}{659}\right)\) | \(e\left(\frac{195}{659}\right)\) | \(e\left(\frac{54}{659}\right)\) | \(e\left(\frac{284}{659}\right)\) | \(e\left(\frac{640}{659}\right)\) | \(e\left(\frac{346}{659}\right)\) | \(e\left(\frac{243}{659}\right)\) |
| \(\chi_{1319}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{333}{659}\right)\) | \(e\left(\frac{281}{659}\right)\) | \(e\left(\frac{7}{659}\right)\) | \(e\left(\frac{601}{659}\right)\) | \(e\left(\frac{614}{659}\right)\) | \(e\left(\frac{241}{659}\right)\) | \(e\left(\frac{340}{659}\right)\) | \(e\left(\frac{562}{659}\right)\) | \(e\left(\frac{275}{659}\right)\) | \(e\left(\frac{96}{659}\right)\) |
| \(\chi_{1319}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{659}\right)\) | \(e\left(\frac{306}{659}\right)\) | \(e\left(\frac{214}{659}\right)\) | \(e\left(\frac{298}{659}\right)\) | \(e\left(\frac{413}{659}\right)\) | \(e\left(\frac{307}{659}\right)\) | \(e\left(\frac{321}{659}\right)\) | \(e\left(\frac{612}{659}\right)\) | \(e\left(\frac{405}{659}\right)\) | \(e\left(\frac{393}{659}\right)\) |
| \(\chi_{1319}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{633}{659}\right)\) | \(e\left(\frac{172}{659}\right)\) | \(e\left(\frac{607}{659}\right)\) | \(e\left(\frac{525}{659}\right)\) | \(e\left(\frac{146}{659}\right)\) | \(e\left(\frac{375}{659}\right)\) | \(e\left(\frac{581}{659}\right)\) | \(e\left(\frac{344}{659}\right)\) | \(e\left(\frac{499}{659}\right)\) | \(e\left(\frac{40}{659}\right)\) |
| \(\chi_{1319}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{659}\right)\) | \(e\left(\frac{383}{659}\right)\) | \(e\left(\frac{298}{659}\right)\) | \(e\left(\frac{261}{659}\right)\) | \(e\left(\frac{532}{659}\right)\) | \(e\left(\frac{563}{659}\right)\) | \(e\left(\frac{447}{659}\right)\) | \(e\left(\frac{107}{659}\right)\) | \(e\left(\frac{410}{659}\right)\) | \(e\left(\frac{227}{659}\right)\) |
| \(\chi_{1319}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{659}\right)\) | \(e\left(\frac{508}{659}\right)\) | \(e\left(\frac{15}{659}\right)\) | \(e\left(\frac{64}{659}\right)\) | \(e\left(\frac{186}{659}\right)\) | \(e\left(\frac{234}{659}\right)\) | \(e\left(\frac{352}{659}\right)\) | \(e\left(\frac{357}{659}\right)\) | \(e\left(\frac{401}{659}\right)\) | \(e\left(\frac{394}{659}\right)\) |
| \(\chi_{1319}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{561}{659}\right)\) | \(e\left(\frac{40}{659}\right)\) | \(e\left(\frac{463}{659}\right)\) | \(e\left(\frac{306}{659}\right)\) | \(e\left(\frac{601}{659}\right)\) | \(e\left(\frac{501}{659}\right)\) | \(e\left(\frac{365}{659}\right)\) | \(e\left(\frac{80}{659}\right)\) | \(e\left(\frac{208}{659}\right)\) | \(e\left(\frac{607}{659}\right)\) |
| \(\chi_{1319}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{156}{659}\right)\) | \(e\left(\frac{286}{659}\right)\) | \(e\left(\frac{312}{659}\right)\) | \(e\left(\frac{145}{659}\right)\) | \(e\left(\frac{442}{659}\right)\) | \(e\left(\frac{386}{659}\right)\) | \(e\left(\frac{468}{659}\right)\) | \(e\left(\frac{572}{659}\right)\) | \(e\left(\frac{301}{659}\right)\) | \(e\left(\frac{419}{659}\right)\) |
| \(\chi_{1319}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{659}\right)\) | \(e\left(\frac{94}{659}\right)\) | \(e\left(\frac{462}{659}\right)\) | \(e\left(\frac{126}{659}\right)\) | \(e\left(\frac{325}{659}\right)\) | \(e\left(\frac{90}{659}\right)\) | \(e\left(\frac{34}{659}\right)\) | \(e\left(\frac{188}{659}\right)\) | \(e\left(\frac{357}{659}\right)\) | \(e\left(\frac{405}{659}\right)\) |
| \(\chi_{1319}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{571}{659}\right)\) | \(e\left(\frac{278}{659}\right)\) | \(e\left(\frac{483}{659}\right)\) | \(e\left(\frac{611}{659}\right)\) | \(e\left(\frac{190}{659}\right)\) | \(e\left(\frac{154}{659}\right)\) | \(e\left(\frac{395}{659}\right)\) | \(e\left(\frac{556}{659}\right)\) | \(e\left(\frac{523}{659}\right)\) | \(e\left(\frac{34}{659}\right)\) |
| \(\chi_{1319}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{468}{659}\right)\) | \(e\left(\frac{199}{659}\right)\) | \(e\left(\frac{277}{659}\right)\) | \(e\left(\frac{435}{659}\right)\) | \(e\left(\frac{8}{659}\right)\) | \(e\left(\frac{499}{659}\right)\) | \(e\left(\frac{86}{659}\right)\) | \(e\left(\frac{398}{659}\right)\) | \(e\left(\frac{244}{659}\right)\) | \(e\left(\frac{598}{659}\right)\) |
| \(\chi_{1319}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{546}{659}\right)\) | \(e\left(\frac{342}{659}\right)\) | \(e\left(\frac{433}{659}\right)\) | \(e\left(\frac{178}{659}\right)\) | \(e\left(\frac{229}{659}\right)\) | \(e\left(\frac{33}{659}\right)\) | \(e\left(\frac{320}{659}\right)\) | \(e\left(\frac{25}{659}\right)\) | \(e\left(\frac{65}{659}\right)\) | \(e\left(\frac{478}{659}\right)\) |
| \(\chi_{1319}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{659}\right)\) | \(e\left(\frac{434}{659}\right)\) | \(e\left(\frac{114}{659}\right)\) | \(e\left(\frac{91}{659}\right)\) | \(e\left(\frac{491}{659}\right)\) | \(e\left(\frac{65}{659}\right)\) | \(e\left(\frac{171}{659}\right)\) | \(e\left(\frac{209}{659}\right)\) | \(e\left(\frac{148}{659}\right)\) | \(e\left(\frac{622}{659}\right)\) |
| \(\chi_{1319}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{565}{659}\right)\) | \(e\left(\frac{267}{659}\right)\) | \(e\left(\frac{471}{659}\right)\) | \(e\left(\frac{428}{659}\right)\) | \(e\left(\frac{173}{659}\right)\) | \(e\left(\frac{494}{659}\right)\) | \(e\left(\frac{377}{659}\right)\) | \(e\left(\frac{534}{659}\right)\) | \(e\left(\frac{334}{659}\right)\) | \(e\left(\frac{246}{659}\right)\) |
| \(\chi_{1319}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{242}{659}\right)\) | \(e\left(\frac{224}{659}\right)\) | \(e\left(\frac{484}{659}\right)\) | \(e\left(\frac{132}{659}\right)\) | \(e\left(\frac{466}{659}\right)\) | \(e\left(\frac{565}{659}\right)\) | \(e\left(\frac{67}{659}\right)\) | \(e\left(\frac{448}{659}\right)\) | \(e\left(\frac{374}{659}\right)\) | \(e\left(\frac{236}{659}\right)\) |
| \(\chi_{1319}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{659}\right)\) | \(e\left(\frac{650}{659}\right)\) | \(e\left(\frac{110}{659}\right)\) | \(e\left(\frac{30}{659}\right)\) | \(e\left(\frac{46}{659}\right)\) | \(e\left(\frac{398}{659}\right)\) | \(e\left(\frac{165}{659}\right)\) | \(e\left(\frac{641}{659}\right)\) | \(e\left(\frac{85}{659}\right)\) | \(e\left(\frac{473}{659}\right)\) |
| \(\chi_{1319}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{659}\right)\) | \(e\left(\frac{6}{659}\right)\) | \(e\left(\frac{366}{659}\right)\) | \(e\left(\frac{639}{659}\right)\) | \(e\left(\frac{189}{659}\right)\) | \(e\left(\frac{174}{659}\right)\) | \(e\left(\frac{549}{659}\right)\) | \(e\left(\frac{12}{659}\right)\) | \(e\left(\frac{163}{659}\right)\) | \(e\left(\frac{124}{659}\right)\) |
| \(\chi_{1319}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{659}\right)\) | \(e\left(\frac{273}{659}\right)\) | \(e\left(\frac{178}{659}\right)\) | \(e\left(\frac{408}{659}\right)\) | \(e\left(\frac{362}{659}\right)\) | \(e\left(\frac{9}{659}\right)\) | \(e\left(\frac{267}{659}\right)\) | \(e\left(\frac{546}{659}\right)\) | \(e\left(\frac{497}{659}\right)\) | \(e\left(\frac{370}{659}\right)\) |
| \(\chi_{1319}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{98}{659}\right)\) | \(e\left(\frac{619}{659}\right)\) | \(e\left(\frac{196}{659}\right)\) | \(e\left(\frac{353}{659}\right)\) | \(e\left(\frac{58}{659}\right)\) | \(e\left(\frac{158}{659}\right)\) | \(e\left(\frac{294}{659}\right)\) | \(e\left(\frac{579}{659}\right)\) | \(e\left(\frac{451}{659}\right)\) | \(e\left(\frac{52}{659}\right)\) |
| \(\chi_{1319}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{659}\right)\) | \(e\left(\frac{617}{659}\right)\) | \(e\left(\frac{74}{659}\right)\) | \(e\left(\frac{140}{659}\right)\) | \(e\left(\frac{654}{659}\right)\) | \(e\left(\frac{100}{659}\right)\) | \(e\left(\frac{111}{659}\right)\) | \(e\left(\frac{575}{659}\right)\) | \(e\left(\frac{177}{659}\right)\) | \(e\left(\frac{450}{659}\right)\) |