from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100315, base_ring=CyclotomicField(20062))
M = H._module
chi = DirichletCharacter(H, M([0,12536]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,100315))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(100315\) | |
| Conductor: | \(20063\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10031\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 20063.g | 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: | $\Q(\zeta_{10031})$ |
| Fixed field: | Number field defined by a degree 10031 polynomial (not computed) |
First 31 of 8592 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{100315}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{728}{1433}\right)\) | \(e\left(\frac{7760}{10031}\right)\) | \(e\left(\frac{23}{1433}\right)\) | \(e\left(\frac{2825}{10031}\right)\) | \(e\left(\frac{4939}{10031}\right)\) | \(e\left(\frac{751}{1433}\right)\) | \(e\left(\frac{5489}{10031}\right)\) | \(e\left(\frac{3659}{10031}\right)\) | \(e\left(\frac{7921}{10031}\right)\) | \(e\left(\frac{6024}{10031}\right)\) |
| \(\chi_{100315}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1263}{1433}\right)\) | \(e\left(\frac{4849}{10031}\right)\) | \(e\left(\frac{1093}{1433}\right)\) | \(e\left(\frac{3659}{10031}\right)\) | \(e\left(\frac{7299}{10031}\right)\) | \(e\left(\frac{923}{1433}\right)\) | \(e\left(\frac{9698}{10031}\right)\) | \(e\left(\frac{8464}{10031}\right)\) | \(e\left(\frac{2469}{10031}\right)\) | \(e\left(\frac{1727}{10031}\right)\) |
| \(\chi_{100315}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{446}{1433}\right)\) | \(e\left(\frac{7998}{10031}\right)\) | \(e\left(\frac{892}{1433}\right)\) | \(e\left(\frac{1089}{10031}\right)\) | \(e\left(\frac{8061}{10031}\right)\) | \(e\left(\frac{1338}{1433}\right)\) | \(e\left(\frac{5965}{10031}\right)\) | \(e\left(\frac{537}{10031}\right)\) | \(e\left(\frac{4211}{10031}\right)\) | \(e\left(\frac{9580}{10031}\right)\) |
| \(\chi_{100315}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1055}{1433}\right)\) | \(e\left(\frac{8978}{10031}\right)\) | \(e\left(\frac{677}{1433}\right)\) | \(e\left(\frac{6332}{10031}\right)\) | \(e\left(\frac{7935}{10031}\right)\) | \(e\left(\frac{299}{1433}\right)\) | \(e\left(\frac{7925}{10031}\right)\) | \(e\left(\frac{663}{10031}\right)\) | \(e\left(\frac{3686}{10031}\right)\) | \(e\left(\frac{620}{10031}\right)\) |
| \(\chi_{100315}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{1433}\right)\) | \(e\left(\frac{5489}{10031}\right)\) | \(e\left(\frac{46}{1433}\right)\) | \(e\left(\frac{5650}{10031}\right)\) | \(e\left(\frac{9878}{10031}\right)\) | \(e\left(\frac{69}{1433}\right)\) | \(e\left(\frac{947}{10031}\right)\) | \(e\left(\frac{7318}{10031}\right)\) | \(e\left(\frac{5811}{10031}\right)\) | \(e\left(\frac{2017}{10031}\right)\) |
| \(\chi_{100315}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{120}{1433}\right)\) | \(e\left(\frac{1972}{10031}\right)\) | \(e\left(\frac{240}{1433}\right)\) | \(e\left(\frac{2812}{10031}\right)\) | \(e\left(\frac{4373}{10031}\right)\) | \(e\left(\frac{360}{1433}\right)\) | \(e\left(\frac{3944}{10031}\right)\) | \(e\left(\frac{1359}{10031}\right)\) | \(e\left(\frac{3652}{10031}\right)\) | \(e\left(\frac{6536}{10031}\right)\) |
| \(\chi_{100315}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{558}{1433}\right)\) | \(e\left(\frac{2578}{10031}\right)\) | \(e\left(\frac{1116}{1433}\right)\) | \(e\left(\frac{6484}{10031}\right)\) | \(e\left(\frac{2207}{10031}\right)\) | \(e\left(\frac{241}{1433}\right)\) | \(e\left(\frac{5156}{10031}\right)\) | \(e\left(\frac{2092}{10031}\right)\) | \(e\left(\frac{359}{10031}\right)\) | \(e\left(\frac{7751}{10031}\right)\) |
| \(\chi_{100315}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1074}{1433}\right)\) | \(e\left(\frac{6472}{10031}\right)\) | \(e\left(\frac{715}{1433}\right)\) | \(e\left(\frac{3959}{10031}\right)\) | \(e\left(\frac{3385}{10031}\right)\) | \(e\left(\frac{356}{1433}\right)\) | \(e\left(\frac{2913}{10031}\right)\) | \(e\left(\frac{5213}{10031}\right)\) | \(e\left(\frac{1446}{10031}\right)\) | \(e\left(\frac{9202}{10031}\right)\) |
| \(\chi_{100315}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1392}{1433}\right)\) | \(e\left(\frac{3673}{10031}\right)\) | \(e\left(\frac{1351}{1433}\right)\) | \(e\left(\frac{3386}{10031}\right)\) | \(e\left(\frac{5444}{10031}\right)\) | \(e\left(\frac{1310}{1433}\right)\) | \(e\left(\frac{7346}{10031}\right)\) | \(e\left(\frac{288}{10031}\right)\) | \(e\left(\frac{3099}{10031}\right)\) | \(e\left(\frac{2448}{10031}\right)\) |
| \(\chi_{100315}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1080}{1433}\right)\) | \(e\left(\frac{3418}{10031}\right)\) | \(e\left(\frac{727}{1433}\right)\) | \(e\left(\frac{947}{10031}\right)\) | \(e\left(\frac{4965}{10031}\right)\) | \(e\left(\frac{374}{1433}\right)\) | \(e\left(\frac{6836}{10031}\right)\) | \(e\left(\frac{9365}{10031}\right)\) | \(e\left(\frac{8507}{10031}\right)\) | \(e\left(\frac{4370}{10031}\right)\) |
| \(\chi_{100315}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1127}{1433}\right)\) | \(e\left(\frac{5289}{10031}\right)\) | \(e\left(\frac{821}{1433}\right)\) | \(e\left(\frac{3147}{10031}\right)\) | \(e\left(\frac{9699}{10031}\right)\) | \(e\left(\frac{515}{1433}\right)\) | \(e\left(\frac{547}{10031}\right)\) | \(e\left(\frac{8930}{10031}\right)\) | \(e\left(\frac{1005}{10031}\right)\) | \(e\left(\frac{5688}{10031}\right)\) |
| \(\chi_{100315}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{198}{1433}\right)\) | \(e\left(\frac{6693}{10031}\right)\) | \(e\left(\frac{396}{1433}\right)\) | \(e\left(\frac{8079}{10031}\right)\) | \(e\left(\frac{9150}{10031}\right)\) | \(e\left(\frac{594}{1433}\right)\) | \(e\left(\frac{3355}{10031}\right)\) | \(e\left(\frac{8046}{10031}\right)\) | \(e\left(\frac{9465}{10031}\right)\) | \(e\left(\frac{8205}{10031}\right)\) |
| \(\chi_{100315}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{894}{1433}\right)\) | \(e\left(\frac{7813}{10031}\right)\) | \(e\left(\frac{355}{1433}\right)\) | \(e\left(\frac{4040}{10031}\right)\) | \(e\left(\frac{6140}{10031}\right)\) | \(e\left(\frac{1249}{1433}\right)\) | \(e\left(\frac{5595}{10031}\right)\) | \(e\left(\frac{1025}{10031}\right)\) | \(e\left(\frac{267}{10031}\right)\) | \(e\left(\frac{3697}{10031}\right)\) |
| \(\chi_{100315}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1349}{1433}\right)\) | \(e\left(\frac{4065}{10031}\right)\) | \(e\left(\frac{1265}{1433}\right)\) | \(e\left(\frac{3477}{10031}\right)\) | \(e\left(\frac{9406}{10031}\right)\) | \(e\left(\frac{1181}{1433}\right)\) | \(e\left(\frac{8130}{10031}\right)\) | \(e\left(\frac{6357}{10031}\right)\) | \(e\left(\frac{2889}{10031}\right)\) | \(e\left(\frac{8895}{10031}\right)\) |
| \(\chi_{100315}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1357}{1433}\right)\) | \(e\left(\frac{2859}{10031}\right)\) | \(e\left(\frac{1281}{1433}\right)\) | \(e\left(\frac{2327}{10031}\right)\) | \(e\left(\frac{1004}{10031}\right)\) | \(e\left(\frac{1205}{1433}\right)\) | \(e\left(\frac{5718}{10031}\right)\) | \(e\left(\frac{429}{10031}\right)\) | \(e\left(\frac{1795}{10031}\right)\) | \(e\left(\frac{8662}{10031}\right)\) |
| \(\chi_{100315}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1093}{1433}\right)\) | \(e\left(\frac{9698}{10031}\right)\) | \(e\left(\frac{753}{1433}\right)\) | \(e\left(\frac{7318}{10031}\right)\) | \(e\left(\frac{4567}{10031}\right)\) | \(e\left(\frac{413}{1433}\right)\) | \(e\left(\frac{9365}{10031}\right)\) | \(e\left(\frac{6897}{10031}\right)\) | \(e\left(\frac{4938}{10031}\right)\) | \(e\left(\frac{3454}{10031}\right)\) |
| \(\chi_{100315}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{308}{1433}\right)\) | \(e\left(\frac{858}{10031}\right)\) | \(e\left(\frac{616}{1433}\right)\) | \(e\left(\frac{3014}{10031}\right)\) | \(e\left(\frac{4680}{10031}\right)\) | \(e\left(\frac{924}{1433}\right)\) | \(e\left(\frac{1716}{10031}\right)\) | \(e\left(\frac{3918}{10031}\right)\) | \(e\left(\frac{5170}{10031}\right)\) | \(e\left(\frac{3210}{10031}\right)\) |
| \(\chi_{100315}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1308}{1433}\right)\) | \(e\left(\frac{9171}{10031}\right)\) | \(e\left(\frac{1183}{1433}\right)\) | \(e\left(\frac{8296}{10031}\right)\) | \(e\left(\frac{6252}{10031}\right)\) | \(e\left(\frac{1058}{1433}\right)\) | \(e\left(\frac{8311}{10031}\right)\) | \(e\left(\frac{913}{10031}\right)\) | \(e\left(\frac{7421}{10031}\right)\) | \(e\left(\frac{2745}{10031}\right)\) |
| \(\chi_{100315}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1174}{1433}\right)\) | \(e\left(\frac{5727}{10031}\right)\) | \(e\left(\frac{915}{1433}\right)\) | \(e\left(\frac{3914}{10031}\right)\) | \(e\left(\frac{2969}{10031}\right)\) | \(e\left(\frac{656}{1433}\right)\) | \(e\left(\frac{1423}{10031}\right)\) | \(e\left(\frac{4196}{10031}\right)\) | \(e\left(\frac{2101}{10031}\right)\) | \(e\left(\frac{5573}{10031}\right)\) |
| \(\chi_{100315}(166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{1433}\right)\) | \(e\left(\frac{415}{10031}\right)\) | \(e\left(\frac{4}{1433}\right)\) | \(e\left(\frac{429}{10031}\right)\) | \(e\left(\frac{8647}{10031}\right)\) | \(e\left(\frac{6}{1433}\right)\) | \(e\left(\frac{830}{10031}\right)\) | \(e\left(\frac{5683}{10031}\right)\) | \(e\left(\frac{443}{10031}\right)\) | \(e\left(\frac{3166}{10031}\right)\) |
| \(\chi_{100315}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1016}{1433}\right)\) | \(e\left(\frac{1602}{10031}\right)\) | \(e\left(\frac{599}{1433}\right)\) | \(e\left(\frac{8714}{10031}\right)\) | \(e\left(\frac{531}{10031}\right)\) | \(e\left(\frac{182}{1433}\right)\) | \(e\left(\frac{3204}{10031}\right)\) | \(e\left(\frac{2335}{10031}\right)\) | \(e\left(\frac{5795}{10031}\right)\) | \(e\left(\frac{4801}{10031}\right)\) |
| \(\chi_{100315}(176,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{1433}\right)\) | \(e\left(\frac{2378}{10031}\right)\) | \(e\left(\frac{458}{1433}\right)\) | \(e\left(\frac{3981}{10031}\right)\) | \(e\left(\frac{2028}{10031}\right)\) | \(e\left(\frac{687}{1433}\right)\) | \(e\left(\frac{4756}{10031}\right)\) | \(e\left(\frac{3704}{10031}\right)\) | \(e\left(\frac{5584}{10031}\right)\) | \(e\left(\frac{1391}{10031}\right)\) |
| \(\chi_{100315}(186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{350}{1433}\right)\) | \(e\left(\frac{6707}{10031}\right)\) | \(e\left(\frac{700}{1433}\right)\) | \(e\left(\frac{9157}{10031}\right)\) | \(e\left(\frac{2843}{10031}\right)\) | \(e\left(\frac{1050}{1433}\right)\) | \(e\left(\frac{3383}{10031}\right)\) | \(e\left(\frac{4322}{10031}\right)\) | \(e\left(\frac{1576}{10031}\right)\) | \(e\left(\frac{6644}{10031}\right)\) |
| \(\chi_{100315}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1256}{1433}\right)\) | \(e\left(\frac{1247}{10031}\right)\) | \(e\left(\frac{1079}{1433}\right)\) | \(e\left(\frac{8}{10031}\right)\) | \(e\left(\frac{4978}{10031}\right)\) | \(e\left(\frac{902}{1433}\right)\) | \(e\left(\frac{2494}{10031}\right)\) | \(e\left(\frac{2187}{10031}\right)\) | \(e\left(\frac{8800}{10031}\right)\) | \(e\left(\frac{3543}{10031}\right)\) |
| \(\chi_{100315}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{953}{1433}\right)\) | \(e\left(\frac{6442}{10031}\right)\) | \(e\left(\frac{473}{1433}\right)\) | \(e\left(\frac{3082}{10031}\right)\) | \(e\left(\frac{6869}{10031}\right)\) | \(e\left(\frac{1426}{1433}\right)\) | \(e\left(\frac{2853}{10031}\right)\) | \(e\left(\frac{7461}{10031}\right)\) | \(e\left(\frac{9753}{10031}\right)\) | \(e\left(\frac{8248}{10031}\right)\) |
| \(\chi_{100315}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{751}{1433}\right)\) | \(e\left(\frac{3218}{10031}\right)\) | \(e\left(\frac{69}{1433}\right)\) | \(e\left(\frac{8475}{10031}\right)\) | \(e\left(\frac{4786}{10031}\right)\) | \(e\left(\frac{820}{1433}\right)\) | \(e\left(\frac{6436}{10031}\right)\) | \(e\left(\frac{946}{10031}\right)\) | \(e\left(\frac{3701}{10031}\right)\) | \(e\left(\frac{8041}{10031}\right)\) |
| \(\chi_{100315}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1042}{1433}\right)\) | \(e\left(\frac{1265}{10031}\right)\) | \(e\left(\frac{651}{1433}\right)\) | \(e\left(\frac{8559}{10031}\right)\) | \(e\left(\frac{6900}{10031}\right)\) | \(e\left(\frac{260}{1433}\right)\) | \(e\left(\frac{2530}{10031}\right)\) | \(e\left(\frac{8863}{10031}\right)\) | \(e\left(\frac{5822}{10031}\right)\) | \(e\left(\frac{103}{10031}\right)\) |
| \(\chi_{100315}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1247}{1433}\right)\) | \(e\left(\frac{96}{10031}\right)\) | \(e\left(\frac{1061}{1433}\right)\) | \(e\left(\frac{8825}{10031}\right)\) | \(e\left(\frac{6907}{10031}\right)\) | \(e\left(\frac{875}{1433}\right)\) | \(e\left(\frac{192}{10031}\right)\) | \(e\left(\frac{8856}{10031}\right)\) | \(e\left(\frac{7523}{10031}\right)\) | \(e\left(\frac{5059}{10031}\right)\) |
| \(\chi_{100315}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{519}{1433}\right)\) | \(e\left(\frac{934}{10031}\right)\) | \(e\left(\frac{1038}{1433}\right)\) | \(e\left(\frac{4567}{10031}\right)\) | \(e\left(\frac{535}{10031}\right)\) | \(e\left(\frac{124}{1433}\right)\) | \(e\left(\frac{1868}{10031}\right)\) | \(e\left(\frac{898}{10031}\right)\) | \(e\left(\frac{8200}{10031}\right)\) | \(e\left(\frac{7633}{10031}\right)\) |
| \(\chi_{100315}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{981}{1433}\right)\) | \(e\left(\frac{788}{10031}\right)\) | \(e\left(\frac{529}{1433}\right)\) | \(e\left(\frac{7655}{10031}\right)\) | \(e\left(\frac{6122}{10031}\right)\) | \(e\left(\frac{77}{1433}\right)\) | \(e\left(\frac{1576}{10031}\right)\) | \(e\left(\frac{2476}{10031}\right)\) | \(e\left(\frac{4491}{10031}\right)\) | \(e\left(\frac{984}{10031}\right)\) |
| \(\chi_{100315}(276,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{848}{1433}\right)\) | \(e\left(\frac{9732}{10031}\right)\) | \(e\left(\frac{263}{1433}\right)\) | \(e\left(\frac{5637}{10031}\right)\) | \(e\left(\frac{9312}{10031}\right)\) | \(e\left(\frac{1111}{1433}\right)\) | \(e\left(\frac{9433}{10031}\right)\) | \(e\left(\frac{5018}{10031}\right)\) | \(e\left(\frac{1542}{10031}\right)\) | \(e\left(\frac{2529}{10031}\right)\) |