from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1163, base_ring=CyclotomicField(1162))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,1163))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1163\) | |
| Conductor: | \(1163\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1162\) | 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: | $\Q(\zeta_{581})$ |
| Fixed field: | Number field defined by a degree 1162 polynomial (not computed) |
First 31 of 492 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1163}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{360}{581}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{1}{1162}\right)\) | \(e\left(\frac{643}{1162}\right)\) | \(e\left(\frac{471}{1162}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{139}{581}\right)\) | \(e\left(\frac{543}{581}\right)\) | \(e\left(\frac{160}{581}\right)\) |
| \(\chi_{1163}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{242}{581}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{643}{1162}\right)\) | \(e\left(\frac{939}{1162}\right)\) | \(e\left(\frac{733}{1162}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{484}{581}\right)\) | \(e\left(\frac{549}{581}\right)\) | \(e\left(\frac{43}{581}\right)\) |
| \(\chi_{1163}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{489}{581}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{471}{1162}\right)\) | \(e\left(\frac{733}{1162}\right)\) | \(e\left(\frac{1061}{1162}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{397}{581}\right)\) | \(e\left(\frac{113}{581}\right)\) | \(e\left(\frac{411}{581}\right)\) |
| \(\chi_{1163}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{394}{581}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{269}{1162}\right)\) | \(e\left(\frac{991}{1162}\right)\) | \(e\left(\frac{41}{1162}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{207}{581}\right)\) | \(e\left(\frac{236}{581}\right)\) | \(e\left(\frac{46}{581}\right)\) |
| \(\chi_{1163}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{367}{581}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{603}{1162}\right)\) | \(e\left(\frac{783}{1162}\right)\) | \(e\left(\frac{485}{1162}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{153}{581}\right)\) | \(e\left(\frac{326}{581}\right)\) | \(e\left(\frac{34}{581}\right)\) |
| \(\chi_{1163}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{316}{581}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{201}{1162}\right)\) | \(e\left(\frac{261}{1162}\right)\) | \(e\left(\frac{549}{1162}\right)\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{51}{581}\right)\) | \(e\left(\frac{496}{581}\right)\) | \(e\left(\frac{205}{581}\right)\) |
| \(\chi_{1163}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{166}\right)\) | \(e\left(\frac{479}{581}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{939}{1162}\right)\) | \(e\left(\frac{699}{1162}\right)\) | \(e\left(\frac{709}{1162}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{377}{581}\right)\) | \(e\left(\frac{340}{581}\right)\) | \(e\left(\frac{342}{581}\right)\) |
| \(\chi_{1163}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{115}{581}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{1009}{1162}\right)\) | \(e\left(\frac{391}{1162}\right)\) | \(e\left(\frac{1143}{1162}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{230}{581}\right)\) | \(e\left(\frac{4}{581}\right)\) | \(e\left(\frac{503}{581}\right)\) |
| \(\chi_{1163}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{166}\right)\) | \(e\left(\frac{563}{581}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{29}{1162}\right)\) | \(e\left(\frac{55}{1162}\right)\) | \(e\left(\frac{877}{1162}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{545}{581}\right)\) | \(e\left(\frac{60}{581}\right)\) | \(e\left(\frac{573}{581}\right)\) |
| \(\chi_{1163}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{330}{581}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{243}{1162}\right)\) | \(e\left(\frac{541}{1162}\right)\) | \(e\left(\frac{577}{1162}\right)\) | \(e\left(\frac{115}{166}\right)\) | \(e\left(\frac{79}{581}\right)\) | \(e\left(\frac{62}{581}\right)\) | \(e\left(\frac{534}{581}\right)\) |
| \(\chi_{1163}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{87}{581}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{925}{1162}\right)\) | \(e\left(\frac{993}{1162}\right)\) | \(e\left(\frac{1087}{1162}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{174}{581}\right)\) | \(e\left(\frac{291}{581}\right)\) | \(e\left(\frac{426}{581}\right)\) |
| \(\chi_{1163}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{578}{581}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{489}{1162}\right)\) | \(e\left(\frac{687}{1162}\right)\) | \(e\left(\frac{243}{1162}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{575}{581}\right)\) | \(e\left(\frac{10}{581}\right)\) | \(e\left(\frac{386}{581}\right)\) |
| \(\chi_{1163}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{244}{581}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{317}{1162}\right)\) | \(e\left(\frac{481}{1162}\right)\) | \(e\left(\frac{571}{1162}\right)\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{488}{581}\right)\) | \(e\left(\frac{155}{581}\right)\) | \(e\left(\frac{173}{581}\right)\) |
| \(\chi_{1163}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{166}\right)\) | \(e\left(\frac{372}{581}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{369}{1162}\right)\) | \(e\left(\frac{219}{1162}\right)\) | \(e\left(\frac{661}{1162}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{163}{581}\right)\) | \(e\left(\frac{503}{581}\right)\) | \(e\left(\frac{359}{581}\right)\) |
| \(\chi_{1163}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{368}{581}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{1021}{1162}\right)\) | \(e\left(\frac{1135}{1162}\right)\) | \(e\left(\frac{985}{1162}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{155}{581}\right)\) | \(e\left(\frac{129}{581}\right)\) | \(e\left(\frac{99}{581}\right)\) |
| \(\chi_{1163}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{468}{581}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{989}{1162}\right)\) | \(e\left(\frac{313}{1162}\right)\) | \(e\left(\frac{1019}{1162}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{355}{581}\right)\) | \(e\left(\frac{183}{581}\right)\) | \(e\left(\frac{208}{581}\right)\) |
| \(\chi_{1163}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{190}{581}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{985}{1162}\right)\) | \(e\left(\frac{65}{1162}\right)\) | \(e\left(\frac{297}{1162}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{380}{581}\right)\) | \(e\left(\frac{335}{581}\right)\) | \(e\left(\frac{149}{581}\right)\) |
| \(\chi_{1163}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{508}{581}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{279}{1162}\right)\) | \(e\left(\frac{449}{1162}\right)\) | \(e\left(\frac{103}{1162}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{435}{581}\right)\) | \(e\left(\frac{437}{581}\right)\) | \(e\left(\frac{484}{581}\right)\) |
| \(\chi_{1163}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{307}{581}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{1087}{1162}\right)\) | \(e\left(\frac{579}{1162}\right)\) | \(e\left(\frac{697}{1162}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{33}{581}\right)\) | \(e\left(\frac{526}{581}\right)\) | \(e\left(\frac{201}{581}\right)\) |
| \(\chi_{1163}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{149}{581}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{115}{1162}\right)\) | \(e\left(\frac{739}{1162}\right)\) | \(e\left(\frac{713}{1162}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{298}{581}\right)\) | \(e\left(\frac{278}{581}\right)\) | \(e\left(\frac{389}{581}\right)\) |
| \(\chi_{1163}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{325}{581}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{477}{1162}\right)\) | \(e\left(\frac{1105}{1162}\right)\) | \(e\left(\frac{401}{1162}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{69}{581}\right)\) | \(e\left(\frac{466}{581}\right)\) | \(e\left(\frac{209}{581}\right)\) |
| \(\chi_{1163}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{390}{581}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{921}{1162}\right)\) | \(e\left(\frac{745}{1162}\right)\) | \(e\left(\frac{365}{1162}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{199}{581}\right)\) | \(e\left(\frac{443}{581}\right)\) | \(e\left(\frac{367}{581}\right)\) |
| \(\chi_{1163}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{522}{581}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{321}{1162}\right)\) | \(e\left(\frac{729}{1162}\right)\) | \(e\left(\frac{131}{1162}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{463}{581}\right)\) | \(e\left(\frac{3}{581}\right)\) | \(e\left(\frac{232}{581}\right)\) |
| \(\chi_{1163}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{404}{581}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{963}{1162}\right)\) | \(e\left(\frac{1025}{1162}\right)\) | \(e\left(\frac{393}{1162}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{227}{581}\right)\) | \(e\left(\frac{9}{581}\right)\) | \(e\left(\frac{115}{581}\right)\) |
| \(\chi_{1163}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{211}{581}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{467}{1162}\right)\) | \(e\left(\frac{485}{1162}\right)\) | \(e\left(\frac{339}{1162}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{422}{581}\right)\) | \(e\left(\frac{265}{581}\right)\) | \(e\left(\frac{352}{581}\right)\) |
| \(\chi_{1163}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{122}{581}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{449}{1162}\right)\) | \(e\left(\frac{531}{1162}\right)\) | \(e\left(\frac{1157}{1162}\right)\) | \(e\left(\frac{123}{166}\right)\) | \(e\left(\frac{244}{581}\right)\) | \(e\left(\frac{368}{581}\right)\) | \(e\left(\frac{377}{581}\right)\) |
| \(\chi_{1163}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{436}{581}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{395}{1162}\right)\) | \(e\left(\frac{669}{1162}\right)\) | \(e\left(\frac{125}{1162}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{291}{581}\right)\) | \(e\left(\frac{96}{581}\right)\) | \(e\left(\frac{452}{581}\right)\) |
| \(\chi_{1163}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{131}{581}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{725}{1162}\right)\) | \(e\left(\frac{213}{1162}\right)\) | \(e\left(\frac{1009}{1162}\right)\) | \(e\left(\frac{145}{166}\right)\) | \(e\left(\frac{262}{581}\right)\) | \(e\left(\frac{338}{581}\right)\) | \(e\left(\frac{381}{581}\right)\) |
| \(\chi_{1163}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{71}{581}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{47}{1162}\right)\) | \(e\left(\frac{9}{1162}\right)\) | \(e\left(\frac{59}{1162}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{142}{581}\right)\) | \(e\left(\frac{538}{581}\right)\) | \(e\left(\frac{548}{581}\right)\) |
| \(\chi_{1163}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{269}{581}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{309}{1162}\right)\) | \(e\left(\frac{1147}{1162}\right)\) | \(e\left(\frac{289}{1162}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{538}{581}\right)\) | \(e\left(\frac{459}{581}\right)\) | \(e\left(\frac{55}{581}\right)\) |
| \(\chi_{1163}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{472}{581}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{337}{1162}\right)\) | \(e\left(\frac{559}{1162}\right)\) | \(e\left(\frac{695}{1162}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{363}{581}\right)\) | \(e\left(\frac{557}{581}\right)\) | \(e\left(\frac{468}{581}\right)\) |