from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6008, base_ring=CyclotomicField(750))
M = H._module
chi = DirichletCharacter(H, M([0,375,577]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,6008))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6008\) | |
| Conductor: | \(6008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(750\) | 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: | $\Q(\zeta_{375})$ |
| Fixed field: | Number field defined by a degree 750 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6008}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{375}\right)\) | \(e\left(\frac{547}{750}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{202}{375}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{241}{750}\right)\) | \(e\left(\frac{749}{750}\right)\) | \(e\left(\frac{83}{750}\right)\) | \(e\left(\frac{511}{750}\right)\) | \(e\left(\frac{79}{750}\right)\) |
| \(\chi_{6008}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{375}\right)\) | \(e\left(\frac{413}{750}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{158}{375}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{389}{750}\right)\) | \(e\left(\frac{571}{750}\right)\) | \(e\left(\frac{607}{750}\right)\) | \(e\left(\frac{719}{750}\right)\) | \(e\left(\frac{641}{750}\right)\) |
| \(\chi_{6008}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{375}\right)\) | \(e\left(\frac{581}{750}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{146}{375}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{293}{750}\right)\) | \(e\left(\frac{727}{750}\right)\) | \(e\left(\frac{409}{750}\right)\) | \(e\left(\frac{503}{750}\right)\) | \(e\left(\frac{317}{750}\right)\) |
| \(\chi_{6008}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{272}{375}\right)\) | \(e\left(\frac{259}{750}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{169}{375}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{727}{750}\right)\) | \(e\left(\frac{53}{750}\right)\) | \(e\left(\frac{101}{750}\right)\) | \(e\left(\frac{667}{750}\right)\) | \(e\left(\frac{313}{750}\right)\) |
| \(\chi_{6008}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{106}{375}\right)\) | \(e\left(\frac{407}{750}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{212}{375}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{71}{750}\right)\) | \(e\left(\frac{619}{750}\right)\) | \(e\left(\frac{373}{750}\right)\) | \(e\left(\frac{191}{750}\right)\) | \(e\left(\frac{599}{750}\right)\) |
| \(\chi_{6008}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{375}\right)\) | \(e\left(\frac{61}{750}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{76}{375}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{733}{750}\right)\) | \(e\left(\frac{137}{750}\right)\) | \(e\left(\frac{629}{750}\right)\) | \(e\left(\frac{493}{750}\right)\) | \(e\left(\frac{427}{750}\right)\) |
| \(\chi_{6008}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{375}\right)\) | \(e\left(\frac{229}{750}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{64}{375}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{637}{750}\right)\) | \(e\left(\frac{293}{750}\right)\) | \(e\left(\frac{431}{750}\right)\) | \(e\left(\frac{277}{750}\right)\) | \(e\left(\frac{103}{750}\right)\) |
| \(\chi_{6008}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{375}\right)\) | \(e\left(\frac{167}{750}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{122}{375}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{101}{750}\right)\) | \(e\left(\frac{289}{750}\right)\) | \(e\left(\frac{13}{750}\right)\) | \(e\left(\frac{71}{750}\right)\) | \(e\left(\frac{419}{750}\right)\) |
| \(\chi_{6008}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{262}{375}\right)\) | \(e\left(\frac{539}{750}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{149}{375}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{317}{750}\right)\) | \(e\left(\frac{313}{750}\right)\) | \(e\left(\frac{271}{750}\right)\) | \(e\left(\frac{557}{750}\right)\) | \(e\left(\frac{23}{750}\right)\) |
| \(\chi_{6008}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{375}\right)\) | \(e\left(\frac{121}{750}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{286}{375}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{163}{750}\right)\) | \(e\left(\frac{407}{750}\right)\) | \(e\left(\frac{719}{750}\right)\) | \(e\left(\frac{523}{750}\right)\) | \(e\left(\frac{97}{750}\right)\) |
| \(\chi_{6008}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{375}\right)\) | \(e\left(\frac{559}{750}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{94}{375}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{127}{750}\right)\) | \(e\left(\frac{653}{750}\right)\) | \(e\left(\frac{551}{750}\right)\) | \(e\left(\frac{67}{750}\right)\) | \(e\left(\frac{163}{750}\right)\) |
| \(\chi_{6008}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{375}\right)\) | \(e\left(\frac{109}{750}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{19}{375}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{277}{750}\right)\) | \(e\left(\frac{503}{750}\right)\) | \(e\left(\frac{251}{750}\right)\) | \(e\left(\frac{217}{750}\right)\) | \(e\left(\frac{13}{750}\right)\) |
| \(\chi_{6008}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{375}\right)\) | \(e\left(\frac{71}{750}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{236}{375}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{263}{750}\right)\) | \(e\left(\frac{307}{750}\right)\) | \(e\left(\frac{19}{750}\right)\) | \(e\left(\frac{623}{750}\right)\) | \(e\left(\frac{497}{750}\right)\) |
| \(\chi_{6008}(381,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{375}\right)\) | \(e\left(\frac{689}{750}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{299}{375}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{17}{750}\right)\) | \(e\left(\frac{613}{750}\right)\) | \(e\left(\frac{121}{750}\right)\) | \(e\left(\frac{257}{750}\right)\) | \(e\left(\frac{323}{750}\right)\) |
| \(\chi_{6008}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{356}{375}\right)\) | \(e\left(\frac{157}{750}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{337}{375}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{571}{750}\right)\) | \(e\left(\frac{119}{750}\right)\) | \(e\left(\frac{623}{750}\right)\) | \(e\left(\frac{691}{750}\right)\) | \(e\left(\frac{349}{750}\right)\) |
| \(\chi_{6008}(413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{375}\right)\) | \(e\left(\frac{577}{750}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{307}{375}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{331}{750}\right)\) | \(e\left(\frac{509}{750}\right)\) | \(e\left(\frac{503}{750}\right)\) | \(e\left(\frac{151}{750}\right)\) | \(e\left(\frac{289}{750}\right)\) |
| \(\chi_{6008}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{184}{375}\right)\) | \(e\left(\frac{473}{750}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{368}{375}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{569}{750}\right)\) | \(e\left(\frac{91}{750}\right)\) | \(e\left(\frac{697}{750}\right)\) | \(e\left(\frac{749}{750}\right)\) | \(e\left(\frac{311}{750}\right)\) |
| \(\chi_{6008}(453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{236}{375}\right)\) | \(e\left(\frac{517}{750}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{97}{375}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{151}{750}\right)\) | \(e\left(\frac{239}{750}\right)\) | \(e\left(\frac{413}{750}\right)\) | \(e\left(\frac{121}{750}\right)\) | \(e\left(\frac{619}{750}\right)\) |
| \(\chi_{6008}(501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{375}\right)\) | \(e\left(\frac{389}{750}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{374}{375}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{617}{750}\right)\) | \(e\left(\frac{13}{750}\right)\) | \(e\left(\frac{421}{750}\right)\) | \(e\left(\frac{107}{750}\right)\) | \(e\left(\frac{473}{750}\right)\) |
| \(\chi_{6008}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{375}\right)\) | \(e\left(\frac{31}{750}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{346}{375}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{643}{750}\right)\) | \(e\left(\frac{377}{750}\right)\) | \(e\left(\frac{209}{750}\right)\) | \(e\left(\frac{103}{750}\right)\) | \(e\left(\frac{217}{750}\right)\) |
| \(\chi_{6008}(517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{88}{375}\right)\) | \(e\left(\frac{161}{750}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{176}{375}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{533}{750}\right)\) | \(e\left(\frac{337}{750}\right)\) | \(e\left(\frac{529}{750}\right)\) | \(e\left(\frac{293}{750}\right)\) | \(e\left(\frac{377}{750}\right)\) |
| \(\chi_{6008}(533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{375}\right)\) | \(e\left(\frac{253}{750}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{223}{375}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{409}{750}\right)\) | \(e\left(\frac{101}{750}\right)\) | \(e\left(\frac{617}{750}\right)\) | \(e\left(\frac{139}{750}\right)\) | \(e\left(\frac{271}{750}\right)\) |
| \(\chi_{6008}(541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{152}{375}\right)\) | \(e\left(\frac{619}{750}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{304}{375}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{307}{750}\right)\) | \(e\left(\frac{173}{750}\right)\) | \(e\left(\frac{641}{750}\right)\) | \(e\left(\frac{97}{750}\right)\) | \(e\left(\frac{583}{750}\right)\) |
| \(\chi_{6008}(565,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{375}\right)\) | \(e\left(\frac{659}{750}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{194}{375}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{677}{750}\right)\) | \(e\left(\frac{103}{750}\right)\) | \(e\left(\frac{451}{750}\right)\) | \(e\left(\frac{617}{750}\right)\) | \(e\left(\frac{113}{750}\right)\) |
| \(\chi_{6008}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{375}\right)\) | \(e\left(\frac{397}{750}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{52}{375}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{541}{750}\right)\) | \(e\left(\frac{449}{750}\right)\) | \(e\left(\frac{233}{750}\right)\) | \(e\left(\frac{61}{750}\right)\) | \(e\left(\frac{529}{750}\right)\) |
| \(\chi_{6008}(629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{375}\right)\) | \(e\left(\frac{211}{750}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{226}{375}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{433}{750}\right)\) | \(e\left(\frac{437}{750}\right)\) | \(e\left(\frac{479}{750}\right)\) | \(e\left(\frac{193}{750}\right)\) | \(e\left(\frac{727}{750}\right)\) |
| \(\chi_{6008}(645,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{238}{375}\right)\) | \(e\left(\frac{461}{750}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{101}{375}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{683}{750}\right)\) | \(e\left(\frac{187}{750}\right)\) | \(e\left(\frac{229}{750}\right)\) | \(e\left(\frac{443}{750}\right)\) | \(e\left(\frac{227}{750}\right)\) |
| \(\chi_{6008}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{375}\right)\) | \(e\left(\frac{323}{750}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{218}{375}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{119}{750}\right)\) | \(e\left(\frac{541}{750}\right)\) | \(e\left(\frac{97}{750}\right)\) | \(e\left(\frac{299}{750}\right)\) | \(e\left(\frac{11}{750}\right)\) |
| \(\chi_{6008}(661,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{202}{375}\right)\) | \(e\left(\frac{719}{750}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{29}{375}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{107}{750}\right)\) | \(e\left(\frac{373}{750}\right)\) | \(e\left(\frac{541}{750}\right)\) | \(e\left(\frac{647}{750}\right)\) | \(e\left(\frac{533}{750}\right)\) |
| \(\chi_{6008}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{344}{375}\right)\) | \(e\left(\frac{493}{750}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{313}{375}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{379}{750}\right)\) | \(e\left(\frac{431}{750}\right)\) | \(e\left(\frac{227}{750}\right)\) | \(e\left(\frac{259}{750}\right)\) | \(e\left(\frac{451}{750}\right)\) |
| \(\chi_{6008}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{375}\right)\) | \(e\left(\frac{277}{750}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{7}{375}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{181}{750}\right)\) | \(e\left(\frac{659}{750}\right)\) | \(e\left(\frac{53}{750}\right)\) | \(e\left(\frac{1}{750}\right)\) | \(e\left(\frac{439}{750}\right)\) |