from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8043, base_ring=CyclotomicField(1146))
M = H._module
chi = DirichletCharacter(H, M([0,382,879]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,8043))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8043\) | |
| Conductor: | \(2681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1146\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2681.p | 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: | $\Q(\zeta_{573})$ |
| Fixed field: | Number field defined by a degree 1146 polynomial (not computed) |
First 31 of 380 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8043}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{256}{573}\right)\) | \(e\left(\frac{512}{573}\right)\) | \(e\left(\frac{497}{1146}\right)\) | \(e\left(\frac{65}{191}\right)\) | \(e\left(\frac{1009}{1146}\right)\) | \(e\left(\frac{547}{1146}\right)\) | \(e\left(\frac{299}{382}\right)\) | \(e\left(\frac{451}{573}\right)\) | \(e\left(\frac{566}{573}\right)\) | \(e\left(\frac{217}{573}\right)\) |
| \(\chi_{8043}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{166}{573}\right)\) | \(e\left(\frac{332}{573}\right)\) | \(e\left(\frac{1043}{1146}\right)\) | \(e\left(\frac{166}{191}\right)\) | \(e\left(\frac{229}{1146}\right)\) | \(e\left(\frac{583}{1146}\right)\) | \(e\left(\frac{267}{382}\right)\) | \(e\left(\frac{91}{573}\right)\) | \(e\left(\frac{179}{573}\right)\) | \(e\left(\frac{181}{573}\right)\) |
| \(\chi_{8043}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{314}{573}\right)\) | \(e\left(\frac{55}{573}\right)\) | \(e\left(\frac{247}{1146}\right)\) | \(e\left(\frac{123}{191}\right)\) | \(e\left(\frac{875}{1146}\right)\) | \(e\left(\frac{371}{1146}\right)\) | \(e\left(\frac{31}{382}\right)\) | \(e\left(\frac{110}{573}\right)\) | \(e\left(\frac{166}{573}\right)\) | \(e\left(\frac{11}{573}\right)\) |
| \(\chi_{8043}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{482}{573}\right)\) | \(e\left(\frac{391}{573}\right)\) | \(e\left(\frac{985}{1146}\right)\) | \(e\left(\frac{100}{191}\right)\) | \(e\left(\frac{803}{1146}\right)\) | \(e\left(\frac{533}{1146}\right)\) | \(e\left(\frac{269}{382}\right)\) | \(e\left(\frac{209}{573}\right)\) | \(e\left(\frac{430}{573}\right)\) | \(e\left(\frac{422}{573}\right)\) |
| \(\chi_{8043}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{104}{573}\right)\) | \(e\left(\frac{208}{573}\right)\) | \(e\left(\frac{757}{1146}\right)\) | \(e\left(\frac{104}{191}\right)\) | \(e\left(\frac{965}{1146}\right)\) | \(e\left(\frac{455}{1146}\right)\) | \(e\left(\frac{211}{382}\right)\) | \(e\left(\frac{416}{573}\right)\) | \(e\left(\frac{409}{573}\right)\) | \(e\left(\frac{500}{573}\right)\) |
| \(\chi_{8043}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{130}{573}\right)\) | \(e\left(\frac{260}{573}\right)\) | \(e\left(\frac{803}{1146}\right)\) | \(e\left(\frac{130}{191}\right)\) | \(e\left(\frac{1063}{1146}\right)\) | \(e\left(\frac{139}{1146}\right)\) | \(e\left(\frac{25}{382}\right)\) | \(e\left(\frac{520}{573}\right)\) | \(e\left(\frac{368}{573}\right)\) | \(e\left(\frac{52}{573}\right)\) |
| \(\chi_{8043}(214,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{573}\right)\) | \(e\left(\frac{121}{573}\right)\) | \(e\left(\frac{85}{1146}\right)\) | \(e\left(\frac{156}{191}\right)\) | \(e\left(\frac{779}{1146}\right)\) | \(e\left(\frac{587}{1146}\right)\) | \(e\left(\frac{221}{382}\right)\) | \(e\left(\frac{242}{573}\right)\) | \(e\left(\frac{136}{573}\right)\) | \(e\left(\frac{368}{573}\right)\) |
| \(\chi_{8043}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{232}{573}\right)\) | \(e\left(\frac{464}{573}\right)\) | \(e\left(\frac{719}{1146}\right)\) | \(e\left(\frac{41}{191}\right)\) | \(e\left(\frac{37}{1146}\right)\) | \(e\left(\frac{1015}{1146}\right)\) | \(e\left(\frac{265}{382}\right)\) | \(e\left(\frac{355}{573}\right)\) | \(e\left(\frac{119}{573}\right)\) | \(e\left(\frac{322}{573}\right)\) |
| \(\chi_{8043}(310,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{142}{573}\right)\) | \(e\left(\frac{284}{573}\right)\) | \(e\left(\frac{119}{1146}\right)\) | \(e\left(\frac{142}{191}\right)\) | \(e\left(\frac{403}{1146}\right)\) | \(e\left(\frac{1051}{1146}\right)\) | \(e\left(\frac{233}{382}\right)\) | \(e\left(\frac{568}{573}\right)\) | \(e\left(\frac{305}{573}\right)\) | \(e\left(\frac{286}{573}\right)\) |
| \(\chi_{8043}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{573}\right)\) | \(e\left(\frac{418}{573}\right)\) | \(e\left(\frac{1075}{1146}\right)\) | \(e\left(\frac{18}{191}\right)\) | \(e\left(\frac{347}{1146}\right)\) | \(e\left(\frac{413}{1146}\right)\) | \(e\left(\frac{121}{382}\right)\) | \(e\left(\frac{263}{573}\right)\) | \(e\left(\frac{1}{573}\right)\) | \(e\left(\frac{542}{573}\right)\) |
| \(\chi_{8043}(340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{573}\right)\) | \(e\left(\frac{478}{573}\right)\) | \(e\left(\frac{511}{1146}\right)\) | \(e\left(\frac{48}{191}\right)\) | \(e\left(\frac{989}{1146}\right)\) | \(e\left(\frac{401}{1146}\right)\) | \(e\left(\frac{259}{382}\right)\) | \(e\left(\frac{383}{573}\right)\) | \(e\left(\frac{130}{573}\right)\) | \(e\left(\frac{554}{573}\right)\) |
| \(\chi_{8043}(352,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{511}{573}\right)\) | \(e\left(\frac{449}{573}\right)\) | \(e\left(\frac{287}{1146}\right)\) | \(e\left(\frac{129}{191}\right)\) | \(e\left(\frac{163}{1146}\right)\) | \(e\left(\frac{445}{1146}\right)\) | \(e\left(\frac{135}{382}\right)\) | \(e\left(\frac{325}{573}\right)\) | \(e\left(\frac{230}{573}\right)\) | \(e\left(\frac{319}{573}\right)\) |
| \(\chi_{8043}(394,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{496}{573}\right)\) | \(e\left(\frac{419}{573}\right)\) | \(e\left(\frac{569}{1146}\right)\) | \(e\left(\frac{114}{191}\right)\) | \(e\left(\frac{415}{1146}\right)\) | \(e\left(\frac{451}{1146}\right)\) | \(e\left(\frac{257}{382}\right)\) | \(e\left(\frac{265}{573}\right)\) | \(e\left(\frac{452}{573}\right)\) | \(e\left(\frac{313}{573}\right)\) |
| \(\chi_{8043}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{398}{573}\right)\) | \(e\left(\frac{223}{573}\right)\) | \(e\left(\frac{43}{1146}\right)\) | \(e\left(\frac{16}{191}\right)\) | \(e\left(\frac{839}{1146}\right)\) | \(e\left(\frac{1025}{1146}\right)\) | \(e\left(\frac{341}{382}\right)\) | \(e\left(\frac{446}{573}\right)\) | \(e\left(\frac{298}{573}\right)\) | \(e\left(\frac{503}{573}\right)\) |
| \(\chi_{8043}(424,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{497}{573}\right)\) | \(e\left(\frac{421}{573}\right)\) | \(e\left(\frac{703}{1146}\right)\) | \(e\left(\frac{115}{191}\right)\) | \(e\left(\frac{551}{1146}\right)\) | \(e\left(\frac{527}{1146}\right)\) | \(e\left(\frac{147}{382}\right)\) | \(e\left(\frac{269}{573}\right)\) | \(e\left(\frac{208}{573}\right)\) | \(e\left(\frac{428}{573}\right)\) |
| \(\chi_{8043}(436,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{106}{573}\right)\) | \(e\left(\frac{212}{573}\right)\) | \(e\left(\frac{1025}{1146}\right)\) | \(e\left(\frac{106}{191}\right)\) | \(e\left(\frac{91}{1146}\right)\) | \(e\left(\frac{607}{1146}\right)\) | \(e\left(\frac{373}{382}\right)\) | \(e\left(\frac{424}{573}\right)\) | \(e\left(\frac{494}{573}\right)\) | \(e\left(\frac{157}{573}\right)\) |
| \(\chi_{8043}(457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{259}{573}\right)\) | \(e\left(\frac{518}{573}\right)\) | \(e\left(\frac{899}{1146}\right)\) | \(e\left(\frac{68}{191}\right)\) | \(e\left(\frac{271}{1146}\right)\) | \(e\left(\frac{775}{1146}\right)\) | \(e\left(\frac{351}{382}\right)\) | \(e\left(\frac{463}{573}\right)\) | \(e\left(\frac{407}{573}\right)\) | \(e\left(\frac{562}{573}\right)\) |
| \(\chi_{8043}(466,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{573}\right)\) | \(e\left(\frac{10}{573}\right)\) | \(e\left(\frac{97}{1146}\right)\) | \(e\left(\frac{5}{191}\right)\) | \(e\left(\frac{107}{1146}\right)\) | \(e\left(\frac{953}{1146}\right)\) | \(e\left(\frac{23}{382}\right)\) | \(e\left(\frac{20}{573}\right)\) | \(e\left(\frac{499}{573}\right)\) | \(e\left(\frac{2}{573}\right)\) |
| \(\chi_{8043}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{355}{573}\right)\) | \(e\left(\frac{137}{573}\right)\) | \(e\left(\frac{11}{1146}\right)\) | \(e\left(\frac{164}{191}\right)\) | \(e\left(\frac{721}{1146}\right)\) | \(e\left(\frac{49}{1146}\right)\) | \(e\left(\frac{105}{382}\right)\) | \(e\left(\frac{274}{573}\right)\) | \(e\left(\frac{476}{573}\right)\) | \(e\left(\frac{142}{573}\right)\) |
| \(\chi_{8043}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{278}{573}\right)\) | \(e\left(\frac{556}{573}\right)\) | \(e\left(\frac{7}{1146}\right)\) | \(e\left(\frac{87}{191}\right)\) | \(e\left(\frac{563}{1146}\right)\) | \(e\left(\frac{1073}{1146}\right)\) | \(e\left(\frac{171}{382}\right)\) | \(e\left(\frac{539}{573}\right)\) | \(e\left(\frac{355}{573}\right)\) | \(e\left(\frac{455}{573}\right)\) |
| \(\chi_{8043}(508,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{573}\right)\) | \(e\left(\frac{442}{573}\right)\) | \(e\left(\frac{391}{1146}\right)\) | \(e\left(\frac{30}{191}\right)\) | \(e\left(\frac{833}{1146}\right)\) | \(e\left(\frac{179}{1146}\right)\) | \(e\left(\frac{329}{382}\right)\) | \(e\left(\frac{311}{573}\right)\) | \(e\left(\frac{511}{573}\right)\) | \(e\left(\frac{203}{573}\right)\) |
| \(\chi_{8043}(541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{573}\right)\) | \(e\left(\frac{338}{573}\right)\) | \(e\left(\frac{299}{1146}\right)\) | \(e\left(\frac{169}{191}\right)\) | \(e\left(\frac{637}{1146}\right)\) | \(e\left(\frac{811}{1146}\right)\) | \(e\left(\frac{319}{382}\right)\) | \(e\left(\frac{103}{573}\right)\) | \(e\left(\frac{20}{573}\right)\) | \(e\left(\frac{526}{573}\right)\) |
| \(\chi_{8043}(550,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{573}\right)\) | \(e\left(\frac{274}{573}\right)\) | \(e\left(\frac{595}{1146}\right)\) | \(e\left(\frac{137}{191}\right)\) | \(e\left(\frac{869}{1146}\right)\) | \(e\left(\frac{671}{1146}\right)\) | \(e\left(\frac{19}{382}\right)\) | \(e\left(\frac{548}{573}\right)\) | \(e\left(\frac{379}{573}\right)\) | \(e\left(\frac{284}{573}\right)\) |
| \(\chi_{8043}(562,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{208}{573}\right)\) | \(e\left(\frac{416}{573}\right)\) | \(e\left(\frac{941}{1146}\right)\) | \(e\left(\frac{17}{191}\right)\) | \(e\left(\frac{211}{1146}\right)\) | \(e\left(\frac{337}{1146}\right)\) | \(e\left(\frac{231}{382}\right)\) | \(e\left(\frac{259}{573}\right)\) | \(e\left(\frac{245}{573}\right)\) | \(e\left(\frac{427}{573}\right)\) |
| \(\chi_{8043}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{449}{573}\right)\) | \(e\left(\frac{325}{573}\right)\) | \(e\left(\frac{1}{1146}\right)\) | \(e\left(\frac{67}{191}\right)\) | \(e\left(\frac{899}{1146}\right)\) | \(e\left(\frac{317}{1146}\right)\) | \(e\left(\frac{79}{382}\right)\) | \(e\left(\frac{77}{573}\right)\) | \(e\left(\frac{460}{573}\right)\) | \(e\left(\frac{65}{573}\right)\) |
| \(\chi_{8043}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{573}\right)\) | \(e\left(\frac{154}{573}\right)\) | \(e\left(\frac{577}{1146}\right)\) | \(e\left(\frac{77}{191}\right)\) | \(e\left(\frac{731}{1146}\right)\) | \(e\left(\frac{695}{1146}\right)\) | \(e\left(\frac{125}{382}\right)\) | \(e\left(\frac{308}{573}\right)\) | \(e\left(\frac{121}{573}\right)\) | \(e\left(\frac{260}{573}\right)\) |
| \(\chi_{8043}(604,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{573}\right)\) | \(e\left(\frac{29}{573}\right)\) | \(e\left(\frac{797}{1146}\right)\) | \(e\left(\frac{110}{191}\right)\) | \(e\left(\frac{253}{1146}\right)\) | \(e\left(\frac{529}{1146}\right)\) | \(e\left(\frac{315}{382}\right)\) | \(e\left(\frac{58}{573}\right)\) | \(e\left(\frac{473}{573}\right)\) | \(e\left(\frac{235}{573}\right)\) |
| \(\chi_{8043}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{573}\right)\) | \(e\left(\frac{553}{573}\right)\) | \(e\left(\frac{379}{1146}\right)\) | \(e\left(\frac{181}{191}\right)\) | \(e\left(\frac{359}{1146}\right)\) | \(e\left(\frac{959}{1146}\right)\) | \(e\left(\frac{145}{382}\right)\) | \(e\left(\frac{533}{573}\right)\) | \(e\left(\frac{148}{573}\right)\) | \(e\left(\frac{569}{573}\right)\) |
| \(\chi_{8043}(697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{338}{573}\right)\) | \(e\left(\frac{103}{573}\right)\) | \(e\left(\frac{25}{1146}\right)\) | \(e\left(\frac{147}{191}\right)\) | \(e\left(\frac{701}{1146}\right)\) | \(e\left(\frac{1049}{1146}\right)\) | \(e\left(\frac{65}{382}\right)\) | \(e\left(\frac{206}{573}\right)\) | \(e\left(\frac{40}{573}\right)\) | \(e\left(\frac{479}{573}\right)\) |
| \(\chi_{8043}(709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{573}\right)\) | \(e\left(\frac{266}{573}\right)\) | \(e\left(\frac{59}{1146}\right)\) | \(e\left(\frac{133}{191}\right)\) | \(e\left(\frac{325}{1146}\right)\) | \(e\left(\frac{367}{1146}\right)\) | \(e\left(\frac{77}{382}\right)\) | \(e\left(\frac{532}{573}\right)\) | \(e\left(\frac{209}{573}\right)\) | \(e\left(\frac{397}{573}\right)\) |
| \(\chi_{8043}(718,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{182}{573}\right)\) | \(e\left(\frac{364}{573}\right)\) | \(e\left(\frac{895}{1146}\right)\) | \(e\left(\frac{182}{191}\right)\) | \(e\left(\frac{113}{1146}\right)\) | \(e\left(\frac{653}{1146}\right)\) | \(e\left(\frac{35}{382}\right)\) | \(e\left(\frac{155}{573}\right)\) | \(e\left(\frac{286}{573}\right)\) | \(e\left(\frac{302}{573}\right)\) |