from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6041, base_ring=CyclotomicField(862))
M = H._module
chi = DirichletCharacter(H, M([431,99]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,6041))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6041\) | |
| Conductor: | \(6041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(862\) | 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: | $\Q(\zeta_{431})$ |
| Fixed field: | Number field defined by a degree 862 polynomial (not computed) |
First 31 of 430 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6041}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{427}{431}\right)\) | \(e\left(\frac{157}{862}\right)\) | \(e\left(\frac{423}{431}\right)\) | \(e\left(\frac{265}{431}\right)\) | \(e\left(\frac{149}{862}\right)\) | \(e\left(\frac{419}{431}\right)\) | \(e\left(\frac{157}{431}\right)\) | \(e\left(\frac{261}{431}\right)\) | \(e\left(\frac{493}{862}\right)\) | \(e\left(\frac{141}{862}\right)\) |
| \(\chi_{6041}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{357}{431}\right)\) | \(e\left(\frac{103}{862}\right)\) | \(e\left(\frac{283}{431}\right)\) | \(e\left(\frac{377}{431}\right)\) | \(e\left(\frac{817}{862}\right)\) | \(e\left(\frac{209}{431}\right)\) | \(e\left(\frac{103}{431}\right)\) | \(e\left(\frac{303}{431}\right)\) | \(e\left(\frac{285}{862}\right)\) | \(e\left(\frac{669}{862}\right)\) |
| \(\chi_{6041}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{270}{431}\right)\) | \(e\left(\frac{393}{862}\right)\) | \(e\left(\frac{109}{431}\right)\) | \(e\left(\frac{430}{431}\right)\) | \(e\left(\frac{71}{862}\right)\) | \(e\left(\frac{379}{431}\right)\) | \(e\left(\frac{393}{431}\right)\) | \(e\left(\frac{269}{431}\right)\) | \(e\left(\frac{125}{862}\right)\) | \(e\left(\frac{611}{862}\right)\) |
| \(\chi_{6041}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{374}{431}\right)\) | \(e\left(\frac{621}{862}\right)\) | \(e\left(\frac{317}{431}\right)\) | \(e\left(\frac{5}{431}\right)\) | \(e\left(\frac{507}{862}\right)\) | \(e\left(\frac{260}{431}\right)\) | \(e\left(\frac{190}{431}\right)\) | \(e\left(\frac{379}{431}\right)\) | \(e\left(\frac{237}{862}\right)\) | \(e\left(\frac{393}{862}\right)\) |
| \(\chi_{6041}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{431}\right)\) | \(e\left(\frac{783}{862}\right)\) | \(e\left(\frac{306}{431}\right)\) | \(e\left(\frac{100}{431}\right)\) | \(e\left(\frac{227}{862}\right)\) | \(e\left(\frac{28}{431}\right)\) | \(e\left(\frac{352}{431}\right)\) | \(e\left(\frac{253}{431}\right)\) | \(e\left(\frac{861}{862}\right)\) | \(e\left(\frac{533}{862}\right)\) |
| \(\chi_{6041}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{431}\right)\) | \(e\left(\frac{201}{862}\right)\) | \(e\left(\frac{138}{431}\right)\) | \(e\left(\frac{62}{431}\right)\) | \(e\left(\frac{339}{862}\right)\) | \(e\left(\frac{207}{431}\right)\) | \(e\left(\frac{201}{431}\right)\) | \(e\left(\frac{131}{431}\right)\) | \(e\left(\frac{439}{862}\right)\) | \(e\left(\frac{477}{862}\right)\) |
| \(\chi_{6041}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{303}{431}\right)\) | \(e\left(\frac{283}{862}\right)\) | \(e\left(\frac{175}{431}\right)\) | \(e\left(\frac{291}{431}\right)\) | \(e\left(\frac{27}{862}\right)\) | \(e\left(\frac{47}{431}\right)\) | \(e\left(\frac{283}{431}\right)\) | \(e\left(\frac{163}{431}\right)\) | \(e\left(\frac{691}{862}\right)\) | \(e\left(\frac{633}{862}\right)\) |
| \(\chi_{6041}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{431}\right)\) | \(e\left(\frac{57}{862}\right)\) | \(e\left(\frac{52}{431}\right)\) | \(e\left(\frac{217}{431}\right)\) | \(e\left(\frac{109}{862}\right)\) | \(e\left(\frac{78}{431}\right)\) | \(e\left(\frac{57}{431}\right)\) | \(e\left(\frac{243}{431}\right)\) | \(e\left(\frac{459}{862}\right)\) | \(e\left(\frac{161}{862}\right)\) |
| \(\chi_{6041}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{431}\right)\) | \(e\left(\frac{335}{862}\right)\) | \(e\left(\frac{230}{431}\right)\) | \(e\left(\frac{247}{431}\right)\) | \(e\left(\frac{565}{862}\right)\) | \(e\left(\frac{345}{431}\right)\) | \(e\left(\frac{335}{431}\right)\) | \(e\left(\frac{362}{431}\right)\) | \(e\left(\frac{157}{862}\right)\) | \(e\left(\frac{795}{862}\right)\) |
| \(\chi_{6041}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{431}\right)\) | \(e\left(\frac{425}{862}\right)\) | \(e\left(\frac{176}{431}\right)\) | \(e\left(\frac{204}{431}\right)\) | \(e\left(\frac{601}{862}\right)\) | \(e\left(\frac{264}{431}\right)\) | \(e\left(\frac{425}{431}\right)\) | \(e\left(\frac{292}{431}\right)\) | \(e\left(\frac{791}{862}\right)\) | \(e\left(\frac{777}{862}\right)\) |
| \(\chi_{6041}(146,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{160}{431}\right)\) | \(e\left(\frac{185}{862}\right)\) | \(e\left(\frac{320}{431}\right)\) | \(e\left(\frac{175}{431}\right)\) | \(e\left(\frac{505}{862}\right)\) | \(e\left(\frac{49}{431}\right)\) | \(e\left(\frac{185}{431}\right)\) | \(e\left(\frac{335}{431}\right)\) | \(e\left(\frac{537}{862}\right)\) | \(e\left(\frac{825}{862}\right)\) |
| \(\chi_{6041}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{431}\right)\) | \(e\left(\frac{229}{862}\right)\) | \(e\left(\frac{35}{431}\right)\) | \(e\left(\frac{403}{431}\right)\) | \(e\left(\frac{695}{862}\right)\) | \(e\left(\frac{268}{431}\right)\) | \(e\left(\frac{229}{431}\right)\) | \(e\left(\frac{205}{431}\right)\) | \(e\left(\frac{483}{862}\right)\) | \(e\left(\frac{299}{862}\right)\) |
| \(\chi_{6041}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{252}{431}\right)\) | \(e\left(\frac{453}{862}\right)\) | \(e\left(\frac{73}{431}\right)\) | \(e\left(\frac{114}{431}\right)\) | \(e\left(\frac{95}{862}\right)\) | \(e\left(\frac{325}{431}\right)\) | \(e\left(\frac{22}{431}\right)\) | \(e\left(\frac{366}{431}\right)\) | \(e\left(\frac{835}{862}\right)\) | \(e\left(\frac{599}{862}\right)\) |
| \(\chi_{6041}(188,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{224}{431}\right)\) | \(e\left(\frac{259}{862}\right)\) | \(e\left(\frac{17}{431}\right)\) | \(e\left(\frac{245}{431}\right)\) | \(e\left(\frac{707}{862}\right)\) | \(e\left(\frac{241}{431}\right)\) | \(e\left(\frac{259}{431}\right)\) | \(e\left(\frac{38}{431}\right)\) | \(e\left(\frac{407}{862}\right)\) | \(e\left(\frac{293}{862}\right)\) |
| \(\chi_{6041}(202,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{268}{431}\right)\) | \(e\left(\frac{687}{862}\right)\) | \(e\left(\frac{105}{431}\right)\) | \(e\left(\frac{347}{431}\right)\) | \(e\left(\frac{361}{862}\right)\) | \(e\left(\frac{373}{431}\right)\) | \(e\left(\frac{256}{431}\right)\) | \(e\left(\frac{184}{431}\right)\) | \(e\left(\frac{587}{862}\right)\) | \(e\left(\frac{35}{862}\right)\) |
| \(\chi_{6041}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{431}\right)\) | \(e\left(\frac{137}{862}\right)\) | \(e\left(\frac{4}{431}\right)\) | \(e\left(\frac{83}{431}\right)\) | \(e\left(\frac{141}{862}\right)\) | \(e\left(\frac{6}{431}\right)\) | \(e\left(\frac{137}{431}\right)\) | \(e\left(\frac{85}{431}\right)\) | \(e\left(\frac{831}{862}\right)\) | \(e\left(\frac{145}{862}\right)\) |
| \(\chi_{6041}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{255}{431}\right)\) | \(e\left(\frac{443}{862}\right)\) | \(e\left(\frac{79}{431}\right)\) | \(e\left(\frac{23}{431}\right)\) | \(e\left(\frac{91}{862}\right)\) | \(e\left(\frac{334}{431}\right)\) | \(e\left(\frac{12}{431}\right)\) | \(e\left(\frac{278}{431}\right)\) | \(e\left(\frac{573}{862}\right)\) | \(e\left(\frac{601}{862}\right)\) |
| \(\chi_{6041}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{431}\right)\) | \(e\left(\frac{21}{862}\right)\) | \(e\left(\frac{246}{431}\right)\) | \(e\left(\frac{148}{431}\right)\) | \(e\left(\frac{267}{862}\right)\) | \(e\left(\frac{369}{431}\right)\) | \(e\left(\frac{21}{431}\right)\) | \(e\left(\frac{271}{431}\right)\) | \(e\left(\frac{33}{862}\right)\) | \(e\left(\frac{513}{862}\right)\) |
| \(\chi_{6041}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{431}\right)\) | \(e\left(\frac{231}{862}\right)\) | \(e\left(\frac{120}{431}\right)\) | \(e\left(\frac{335}{431}\right)\) | \(e\left(\frac{351}{862}\right)\) | \(e\left(\frac{180}{431}\right)\) | \(e\left(\frac{231}{431}\right)\) | \(e\left(\frac{395}{431}\right)\) | \(e\left(\frac{363}{862}\right)\) | \(e\left(\frac{471}{862}\right)\) |
| \(\chi_{6041}(314,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{431}\right)\) | \(e\left(\frac{583}{862}\right)\) | \(e\left(\frac{426}{431}\right)\) | \(e\left(\frac{4}{431}\right)\) | \(e\left(\frac{147}{862}\right)\) | \(e\left(\frac{208}{431}\right)\) | \(e\left(\frac{152}{431}\right)\) | \(e\left(\frac{217}{431}\right)\) | \(e\left(\frac{793}{862}\right)\) | \(e\left(\frac{573}{862}\right)\) |
| \(\chi_{6041}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{431}\right)\) | \(e\left(\frac{659}{862}\right)\) | \(e\left(\frac{208}{431}\right)\) | \(e\left(\frac{6}{431}\right)\) | \(e\left(\frac{5}{862}\right)\) | \(e\left(\frac{312}{431}\right)\) | \(e\left(\frac{228}{431}\right)\) | \(e\left(\frac{110}{431}\right)\) | \(e\left(\frac{543}{862}\right)\) | \(e\left(\frac{213}{862}\right)\) |
| \(\chi_{6041}(356,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{431}\right)\) | \(e\left(\frac{675}{862}\right)\) | \(e\left(\frac{26}{431}\right)\) | \(e\left(\frac{324}{431}\right)\) | \(e\left(\frac{701}{862}\right)\) | \(e\left(\frac{39}{431}\right)\) | \(e\left(\frac{244}{431}\right)\) | \(e\left(\frac{337}{431}\right)\) | \(e\left(\frac{445}{862}\right)\) | \(e\left(\frac{727}{862}\right)\) |
| \(\chi_{6041}(370,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{375}{431}\right)\) | \(e\left(\frac{43}{862}\right)\) | \(e\left(\frac{319}{431}\right)\) | \(e\left(\frac{262}{431}\right)\) | \(e\left(\frac{793}{862}\right)\) | \(e\left(\frac{263}{431}\right)\) | \(e\left(\frac{43}{431}\right)\) | \(e\left(\frac{206}{431}\right)\) | \(e\left(\frac{437}{862}\right)\) | \(e\left(\frac{681}{862}\right)\) |
| \(\chi_{6041}(377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{351}{431}\right)\) | \(e\left(\frac{123}{862}\right)\) | \(e\left(\frac{271}{431}\right)\) | \(e\left(\frac{128}{431}\right)\) | \(e\left(\frac{825}{862}\right)\) | \(e\left(\frac{191}{431}\right)\) | \(e\left(\frac{123}{431}\right)\) | \(e\left(\frac{48}{431}\right)\) | \(e\left(\frac{809}{862}\right)\) | \(e\left(\frac{665}{862}\right)\) |
| \(\chi_{6041}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{431}\right)\) | \(e\left(\frac{109}{862}\right)\) | \(e\left(\frac{107}{431}\right)\) | \(e\left(\frac{173}{431}\right)\) | \(e\left(\frac{647}{862}\right)\) | \(e\left(\frac{376}{431}\right)\) | \(e\left(\frac{109}{431}\right)\) | \(e\left(\frac{11}{431}\right)\) | \(e\left(\frac{787}{862}\right)\) | \(e\left(\frac{323}{862}\right)\) |
| \(\chi_{6041}(405,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{380}{431}\right)\) | \(e\left(\frac{601}{862}\right)\) | \(e\left(\frac{329}{431}\right)\) | \(e\left(\frac{254}{431}\right)\) | \(e\left(\frac{499}{862}\right)\) | \(e\left(\frac{278}{431}\right)\) | \(e\left(\frac{170}{431}\right)\) | \(e\left(\frac{203}{431}\right)\) | \(e\left(\frac{575}{862}\right)\) | \(e\left(\frac{397}{862}\right)\) |
| \(\chi_{6041}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{346}{431}\right)\) | \(e\left(\frac{427}{862}\right)\) | \(e\left(\frac{261}{431}\right)\) | \(e\left(\frac{136}{431}\right)\) | \(e\left(\frac{257}{862}\right)\) | \(e\left(\frac{176}{431}\right)\) | \(e\left(\frac{427}{431}\right)\) | \(e\left(\frac{51}{431}\right)\) | \(e\left(\frac{671}{862}\right)\) | \(e\left(\frac{87}{862}\right)\) |
| \(\chi_{6041}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{431}\right)\) | \(e\left(\frac{699}{862}\right)\) | \(e\left(\frac{184}{431}\right)\) | \(e\left(\frac{370}{431}\right)\) | \(e\left(\frac{21}{862}\right)\) | \(e\left(\frac{276}{431}\right)\) | \(e\left(\frac{268}{431}\right)\) | \(e\left(\frac{31}{431}\right)\) | \(e\left(\frac{729}{862}\right)\) | \(e\left(\frac{205}{862}\right)\) |
| \(\chi_{6041}(468,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{431}\right)\) | \(e\left(\frac{101}{862}\right)\) | \(e\left(\frac{198}{431}\right)\) | \(e\left(\frac{14}{431}\right)\) | \(e\left(\frac{299}{862}\right)\) | \(e\left(\frac{297}{431}\right)\) | \(e\left(\frac{101}{431}\right)\) | \(e\left(\frac{113}{431}\right)\) | \(e\left(\frac{405}{862}\right)\) | \(e\left(\frac{497}{862}\right)\) |
| \(\chi_{6041}(482,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{431}\right)\) | \(e\left(\frac{603}{862}\right)\) | \(e\left(\frac{414}{431}\right)\) | \(e\left(\frac{186}{431}\right)\) | \(e\left(\frac{155}{862}\right)\) | \(e\left(\frac{190}{431}\right)\) | \(e\left(\frac{172}{431}\right)\) | \(e\left(\frac{393}{431}\right)\) | \(e\left(\frac{455}{862}\right)\) | \(e\left(\frac{569}{862}\right)\) |
| \(\chi_{6041}(510,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{431}\right)\) | \(e\left(\frac{499}{862}\right)\) | \(e\left(\frac{304}{431}\right)\) | \(e\left(\frac{274}{431}\right)\) | \(e\left(\frac{803}{862}\right)\) | \(e\left(\frac{25}{431}\right)\) | \(e\left(\frac{68}{431}\right)\) | \(e\left(\frac{426}{431}\right)\) | \(e\left(\frac{661}{862}\right)\) | \(e\left(\frac{245}{862}\right)\) |