from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8044, base_ring=CyclotomicField(2010))
M = H._module
chi = DirichletCharacter(H, M([1005,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,8044))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8044\) | |
| Conductor: | \(8044\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2010\) | 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: | $\Q(\zeta_{1005})$ |
| Fixed field: | Number field defined by a degree 2010 polynomial (not computed) |
First 31 of 528 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8044}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{503}{1005}\right)\) | \(e\left(\frac{512}{1005}\right)\) | \(e\left(\frac{401}{1005}\right)\) | \(e\left(\frac{1}{1005}\right)\) | \(e\left(\frac{16}{1005}\right)\) | \(e\left(\frac{323}{335}\right)\) | \(e\left(\frac{2}{201}\right)\) | \(e\left(\frac{347}{2010}\right)\) | \(e\left(\frac{79}{1005}\right)\) | \(e\left(\frac{904}{1005}\right)\) |
| \(\chi_{8044}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{1005}\right)\) | \(e\left(\frac{584}{1005}\right)\) | \(e\left(\frac{2}{1005}\right)\) | \(e\left(\frac{802}{1005}\right)\) | \(e\left(\frac{772}{1005}\right)\) | \(e\left(\frac{91}{335}\right)\) | \(e\left(\frac{197}{201}\right)\) | \(e\left(\frac{1919}{2010}\right)\) | \(e\left(\frac{43}{1005}\right)\) | \(e\left(\frac{403}{1005}\right)\) |
| \(\chi_{8044}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{1005}\right)\) | \(e\left(\frac{304}{1005}\right)\) | \(e\left(\frac{772}{1005}\right)\) | \(e\left(\frac{32}{1005}\right)\) | \(e\left(\frac{512}{1005}\right)\) | \(e\left(\frac{286}{335}\right)\) | \(e\left(\frac{64}{201}\right)\) | \(e\left(\frac{49}{2010}\right)\) | \(e\left(\frac{518}{1005}\right)\) | \(e\left(\frac{788}{1005}\right)\) |
| \(\chi_{8044}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{1005}\right)\) | \(e\left(\frac{496}{1005}\right)\) | \(e\left(\frac{43}{1005}\right)\) | \(e\left(\frac{158}{1005}\right)\) | \(e\left(\frac{518}{1005}\right)\) | \(e\left(\frac{114}{335}\right)\) | \(e\left(\frac{115}{201}\right)\) | \(e\left(\frac{1561}{2010}\right)\) | \(e\left(\frac{422}{1005}\right)\) | \(e\left(\frac{122}{1005}\right)\) |
| \(\chi_{8044}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{913}{1005}\right)\) | \(e\left(\frac{262}{1005}\right)\) | \(e\left(\frac{586}{1005}\right)\) | \(e\left(\frac{821}{1005}\right)\) | \(e\left(\frac{71}{1005}\right)\) | \(e\left(\frac{198}{335}\right)\) | \(e\left(\frac{34}{201}\right)\) | \(e\left(\frac{1477}{2010}\right)\) | \(e\left(\frac{539}{1005}\right)\) | \(e\left(\frac{494}{1005}\right)\) |
| \(\chi_{8044}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{1005}\right)\) | \(e\left(\frac{833}{1005}\right)\) | \(e\left(\frac{674}{1005}\right)\) | \(e\left(\frac{934}{1005}\right)\) | \(e\left(\frac{874}{1005}\right)\) | \(e\left(\frac{182}{335}\right)\) | \(e\left(\frac{59}{201}\right)\) | \(e\left(\frac{1493}{2010}\right)\) | \(e\left(\frac{421}{1005}\right)\) | \(e\left(\frac{136}{1005}\right)\) |
| \(\chi_{8044}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{1005}\right)\) | \(e\left(\frac{293}{1005}\right)\) | \(e\left(\frac{149}{1005}\right)\) | \(e\left(\frac{454}{1005}\right)\) | \(e\left(\frac{229}{1005}\right)\) | \(e\left(\frac{247}{335}\right)\) | \(e\left(\frac{104}{201}\right)\) | \(e\left(\frac{1763}{2010}\right)\) | \(e\left(\frac{691}{1005}\right)\) | \(e\left(\frac{376}{1005}\right)\) |
| \(\chi_{8044}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{1005}\right)\) | \(e\left(\frac{323}{1005}\right)\) | \(e\left(\frac{569}{1005}\right)\) | \(e\left(\frac{34}{1005}\right)\) | \(e\left(\frac{544}{1005}\right)\) | \(e\left(\frac{262}{335}\right)\) | \(e\left(\frac{68}{201}\right)\) | \(e\left(\frac{743}{2010}\right)\) | \(e\left(\frac{676}{1005}\right)\) | \(e\left(\frac{586}{1005}\right)\) |
| \(\chi_{8044}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{562}{1005}\right)\) | \(e\left(\frac{628}{1005}\right)\) | \(e\left(\frac{484}{1005}\right)\) | \(e\left(\frac{119}{1005}\right)\) | \(e\left(\frac{899}{1005}\right)\) | \(e\left(\frac{247}{335}\right)\) | \(e\left(\frac{37}{201}\right)\) | \(e\left(\frac{1093}{2010}\right)\) | \(e\left(\frac{356}{1005}\right)\) | \(e\left(\frac{41}{1005}\right)\) |
| \(\chi_{8044}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{326}{1005}\right)\) | \(e\left(\frac{164}{1005}\right)\) | \(e\left(\frac{152}{1005}\right)\) | \(e\left(\frac{652}{1005}\right)\) | \(e\left(\frac{382}{1005}\right)\) | \(e\left(\frac{216}{335}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{119}{2010}\right)\) | \(e\left(\frac{253}{1005}\right)\) | \(e\left(\frac{478}{1005}\right)\) |
| \(\chi_{8044}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{698}{1005}\right)\) | \(e\left(\frac{197}{1005}\right)\) | \(e\left(\frac{11}{1005}\right)\) | \(e\left(\frac{391}{1005}\right)\) | \(e\left(\frac{226}{1005}\right)\) | \(e\left(\frac{333}{335}\right)\) | \(e\left(\frac{179}{201}\right)\) | \(e\left(\frac{1007}{2010}\right)\) | \(e\left(\frac{739}{1005}\right)\) | \(e\left(\frac{709}{1005}\right)\) |
| \(\chi_{8044}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{1005}\right)\) | \(e\left(\frac{209}{1005}\right)\) | \(e\left(\frac{782}{1005}\right)\) | \(e\left(\frac{22}{1005}\right)\) | \(e\left(\frac{352}{1005}\right)\) | \(e\left(\frac{71}{335}\right)\) | \(e\left(\frac{44}{201}\right)\) | \(e\left(\frac{599}{2010}\right)\) | \(e\left(\frac{733}{1005}\right)\) | \(e\left(\frac{793}{1005}\right)\) |
| \(\chi_{8044}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{658}{1005}\right)\) | \(e\left(\frac{442}{1005}\right)\) | \(e\left(\frac{91}{1005}\right)\) | \(e\left(\frac{311}{1005}\right)\) | \(e\left(\frac{956}{1005}\right)\) | \(e\left(\frac{288}{335}\right)\) | \(e\left(\frac{19}{201}\right)\) | \(e\left(\frac{1387}{2010}\right)\) | \(e\left(\frac{449}{1005}\right)\) | \(e\left(\frac{749}{1005}\right)\) |
| \(\chi_{8044}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{508}{1005}\right)\) | \(e\left(\frac{607}{1005}\right)\) | \(e\left(\frac{391}{1005}\right)\) | \(e\left(\frac{11}{1005}\right)\) | \(e\left(\frac{176}{1005}\right)\) | \(e\left(\frac{203}{335}\right)\) | \(e\left(\frac{22}{201}\right)\) | \(e\left(\frac{1807}{2010}\right)\) | \(e\left(\frac{869}{1005}\right)\) | \(e\left(\frac{899}{1005}\right)\) |
| \(\chi_{8044}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{979}{1005}\right)\) | \(e\left(\frac{511}{1005}\right)\) | \(e\left(\frac{253}{1005}\right)\) | \(e\left(\frac{953}{1005}\right)\) | \(e\left(\frac{173}{1005}\right)\) | \(e\left(\frac{289}{335}\right)\) | \(e\left(\frac{97}{201}\right)\) | \(e\left(\frac{1051}{2010}\right)\) | \(e\left(\frac{917}{1005}\right)\) | \(e\left(\frac{227}{1005}\right)\) |
| \(\chi_{8044}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{244}{1005}\right)\) | \(e\left(\frac{616}{1005}\right)\) | \(e\left(\frac{718}{1005}\right)\) | \(e\left(\frac{488}{1005}\right)\) | \(e\left(\frac{773}{1005}\right)\) | \(e\left(\frac{174}{335}\right)\) | \(e\left(\frac{172}{201}\right)\) | \(e\left(\frac{1501}{2010}\right)\) | \(e\left(\frac{362}{1005}\right)\) | \(e\left(\frac{962}{1005}\right)\) |
| \(\chi_{8044}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{1005}\right)\) | \(e\left(\frac{61}{1005}\right)\) | \(e\left(\frac{988}{1005}\right)\) | \(e\left(\frac{218}{1005}\right)\) | \(e\left(\frac{473}{1005}\right)\) | \(e\left(\frac{64}{335}\right)\) | \(e\left(\frac{34}{201}\right)\) | \(e\left(\frac{271}{2010}\right)\) | \(e\left(\frac{137}{1005}\right)\) | \(e\left(\frac{92}{1005}\right)\) |
| \(\chi_{8044}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{833}{1005}\right)\) | \(e\left(\frac{752}{1005}\right)\) | \(e\left(\frac{746}{1005}\right)\) | \(e\left(\frac{661}{1005}\right)\) | \(e\left(\frac{526}{1005}\right)\) | \(e\left(\frac{108}{335}\right)\) | \(e\left(\frac{116}{201}\right)\) | \(e\left(\frac{227}{2010}\right)\) | \(e\left(\frac{964}{1005}\right)\) | \(e\left(\frac{574}{1005}\right)\) |
| \(\chi_{8044}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{1005}\right)\) | \(e\left(\frac{878}{1005}\right)\) | \(e\left(\frac{299}{1005}\right)\) | \(e\left(\frac{304}{1005}\right)\) | \(e\left(\frac{844}{1005}\right)\) | \(e\left(\frac{37}{335}\right)\) | \(e\left(\frac{5}{201}\right)\) | \(e\left(\frac{1973}{2010}\right)\) | \(e\left(\frac{901}{1005}\right)\) | \(e\left(\frac{451}{1005}\right)\) |
| \(\chi_{8044}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{1005}\right)\) | \(e\left(\frac{817}{1005}\right)\) | \(e\left(\frac{316}{1005}\right)\) | \(e\left(\frac{86}{1005}\right)\) | \(e\left(\frac{371}{1005}\right)\) | \(e\left(\frac{308}{335}\right)\) | \(e\left(\frac{172}{201}\right)\) | \(e\left(\frac{697}{2010}\right)\) | \(e\left(\frac{764}{1005}\right)\) | \(e\left(\frac{359}{1005}\right)\) |
| \(\chi_{8044}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{638}{1005}\right)\) | \(e\left(\frac{62}{1005}\right)\) | \(e\left(\frac{131}{1005}\right)\) | \(e\left(\frac{271}{1005}\right)\) | \(e\left(\frac{316}{1005}\right)\) | \(e\left(\frac{98}{335}\right)\) | \(e\left(\frac{140}{201}\right)\) | \(e\left(\frac{1577}{2010}\right)\) | \(e\left(\frac{304}{1005}\right)\) | \(e\left(\frac{769}{1005}\right)\) |
| \(\chi_{8044}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{806}{1005}\right)\) | \(e\left(\frac{239}{1005}\right)\) | \(e\left(\frac{197}{1005}\right)\) | \(e\left(\frac{607}{1005}\right)\) | \(e\left(\frac{667}{1005}\right)\) | \(e\left(\frac{86}{335}\right)\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{1589}{2010}\right)\) | \(e\left(\frac{718}{1005}\right)\) | \(e\left(\frac{1003}{1005}\right)\) |
| \(\chi_{8044}(287,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{224}{1005}\right)\) | \(e\left(\frac{236}{1005}\right)\) | \(e\left(\frac{758}{1005}\right)\) | \(e\left(\frac{448}{1005}\right)\) | \(e\left(\frac{133}{1005}\right)\) | \(e\left(\frac{319}{335}\right)\) | \(e\left(\frac{92}{201}\right)\) | \(e\left(\frac{1691}{2010}\right)\) | \(e\left(\frac{217}{1005}\right)\) | \(e\left(\frac{982}{1005}\right)\) |
| \(\chi_{8044}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{728}{1005}\right)\) | \(e\left(\frac{767}{1005}\right)\) | \(e\left(\frac{956}{1005}\right)\) | \(e\left(\frac{451}{1005}\right)\) | \(e\left(\frac{181}{1005}\right)\) | \(e\left(\frac{283}{335}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{1727}{2010}\right)\) | \(e\left(\frac{454}{1005}\right)\) | \(e\left(\frac{679}{1005}\right)\) |
| \(\chi_{8044}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{752}{1005}\right)\) | \(e\left(\frac{218}{1005}\right)\) | \(e\left(\frac{104}{1005}\right)\) | \(e\left(\frac{499}{1005}\right)\) | \(e\left(\frac{949}{1005}\right)\) | \(e\left(\frac{42}{335}\right)\) | \(e\left(\frac{194}{201}\right)\) | \(e\left(\frac{293}{2010}\right)\) | \(e\left(\frac{226}{1005}\right)\) | \(e\left(\frac{856}{1005}\right)\) |
| \(\chi_{8044}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{914}{1005}\right)\) | \(e\left(\frac{281}{1005}\right)\) | \(e\left(\frac{383}{1005}\right)\) | \(e\left(\frac{823}{1005}\right)\) | \(e\left(\frac{103}{1005}\right)\) | \(e\left(\frac{174}{335}\right)\) | \(e\left(\frac{38}{201}\right)\) | \(e\left(\frac{161}{2010}\right)\) | \(e\left(\frac{697}{1005}\right)\) | \(e\left(\frac{292}{1005}\right)\) |
| \(\chi_{8044}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{679}{1005}\right)\) | \(e\left(\frac{841}{1005}\right)\) | \(e\left(\frac{853}{1005}\right)\) | \(e\left(\frac{353}{1005}\right)\) | \(e\left(\frac{623}{1005}\right)\) | \(e\left(\frac{119}{335}\right)\) | \(e\left(\frac{103}{201}\right)\) | \(e\left(\frac{1891}{2010}\right)\) | \(e\left(\frac{752}{1005}\right)\) | \(e\left(\frac{527}{1005}\right)\) |
| \(\chi_{8044}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{901}{1005}\right)\) | \(e\left(\frac{34}{1005}\right)\) | \(e\left(\frac{7}{1005}\right)\) | \(e\left(\frac{797}{1005}\right)\) | \(e\left(\frac{692}{1005}\right)\) | \(e\left(\frac{151}{335}\right)\) | \(e\left(\frac{187}{201}\right)\) | \(e\left(\frac{1189}{2010}\right)\) | \(e\left(\frac{653}{1005}\right)\) | \(e\left(\frac{908}{1005}\right)\) |
| \(\chi_{8044}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{862}{1005}\right)\) | \(e\left(\frac{298}{1005}\right)\) | \(e\left(\frac{889}{1005}\right)\) | \(e\left(\frac{719}{1005}\right)\) | \(e\left(\frac{449}{1005}\right)\) | \(e\left(\frac{82}{335}\right)\) | \(e\left(\frac{31}{201}\right)\) | \(e\left(\frac{253}{2010}\right)\) | \(e\left(\frac{521}{1005}\right)\) | \(e\left(\frac{746}{1005}\right)\) |
| \(\chi_{8044}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{1005}\right)\) | \(e\left(\frac{551}{1005}\right)\) | \(e\left(\frac{143}{1005}\right)\) | \(e\left(\frac{58}{1005}\right)\) | \(e\left(\frac{928}{1005}\right)\) | \(e\left(\frac{309}{335}\right)\) | \(e\left(\frac{116}{201}\right)\) | \(e\left(\frac{1031}{2010}\right)\) | \(e\left(\frac{562}{1005}\right)\) | \(e\left(\frac{172}{1005}\right)\) |
| \(\chi_{8044}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{1005}\right)\) | \(e\left(\frac{914}{1005}\right)\) | \(e\left(\frac{602}{1005}\right)\) | \(e\left(\frac{202}{1005}\right)\) | \(e\left(\frac{217}{1005}\right)\) | \(e\left(\frac{256}{335}\right)\) | \(e\left(\frac{2}{201}\right)\) | \(e\left(\frac{749}{2010}\right)\) | \(e\left(\frac{883}{1005}\right)\) | \(e\left(\frac{703}{1005}\right)\) |