from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6009, base_ring=CyclotomicField(2002))
M = H._module
chi = DirichletCharacter(H, M([1001,1460]))
chi.galois_orbit()
[g,chi] = znchar(Mod(14,6009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6009\) | |
| Conductor: | \(6009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2002\) | 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: | $\Q(\zeta_{1001})$ |
| Fixed field: | Number field defined by a degree 2002 polynomial (not computed) |
First 31 of 720 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6009}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{286}\right)\) | \(e\left(\frac{10}{143}\right)\) | \(e\left(\frac{459}{2002}\right)\) | \(e\left(\frac{701}{1001}\right)\) | \(e\left(\frac{173}{286}\right)\) | \(e\left(\frac{765}{1001}\right)\) | \(e\left(\frac{67}{286}\right)\) | \(e\left(\frac{531}{1001}\right)\) | \(e\left(\frac{471}{2002}\right)\) | \(e\left(\frac{20}{143}\right)\) |
| \(\chi_{6009}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{286}\right)\) | \(e\left(\frac{89}{143}\right)\) | \(e\left(\frac{1125}{2002}\right)\) | \(e\left(\frac{619}{1001}\right)\) | \(e\left(\frac{267}{286}\right)\) | \(e\left(\frac{874}{1001}\right)\) | \(e\left(\frac{153}{286}\right)\) | \(e\left(\frac{536}{1001}\right)\) | \(e\left(\frac{1861}{2002}\right)\) | \(e\left(\frac{35}{143}\right)\) |
| \(\chi_{6009}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{286}\right)\) | \(e\left(\frac{115}{143}\right)\) | \(e\left(\frac{1775}{2002}\right)\) | \(e\left(\frac{554}{1001}\right)\) | \(e\left(\frac{59}{286}\right)\) | \(e\left(\frac{289}{1001}\right)\) | \(e\left(\frac{127}{286}\right)\) | \(e\left(\frac{601}{1001}\right)\) | \(e\left(\frac{1913}{2002}\right)\) | \(e\left(\frac{87}{143}\right)\) |
| \(\chi_{6009}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{286}\right)\) | \(e\left(\frac{7}{143}\right)\) | \(e\left(\frac{1737}{2002}\right)\) | \(e\left(\frac{219}{1001}\right)\) | \(e\left(\frac{21}{286}\right)\) | \(e\left(\frac{893}{1001}\right)\) | \(e\left(\frac{147}{286}\right)\) | \(e\left(\frac{243}{1001}\right)\) | \(e\left(\frac{487}{2002}\right)\) | \(e\left(\frac{14}{143}\right)\) |
| \(\chi_{6009}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{286}\right)\) | \(e\left(\frac{96}{143}\right)\) | \(e\left(\frac{1861}{2002}\right)\) | \(e\left(\frac{838}{1001}\right)\) | \(e\left(\frac{145}{286}\right)\) | \(e\left(\frac{766}{1001}\right)\) | \(e\left(\frac{157}{286}\right)\) | \(e\left(\frac{779}{1001}\right)\) | \(e\left(\frac{1347}{2002}\right)\) | \(e\left(\frac{49}{143}\right)\) |
| \(\chi_{6009}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{286}\right)\) | \(e\left(\frac{131}{143}\right)\) | \(e\left(\frac{965}{2002}\right)\) | \(e\left(\frac{789}{1001}\right)\) | \(e\left(\frac{107}{286}\right)\) | \(e\left(\frac{941}{1001}\right)\) | \(e\left(\frac{177}{286}\right)\) | \(e\left(\frac{135}{1001}\right)\) | \(e\left(\frac{493}{2002}\right)\) | \(e\left(\frac{119}{143}\right)\) |
| \(\chi_{6009}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{286}\right)\) | \(e\left(\frac{105}{143}\right)\) | \(e\left(\frac{1173}{2002}\right)\) | \(e\left(\frac{568}{1001}\right)\) | \(e\left(\frac{29}{286}\right)\) | \(e\left(\frac{954}{1001}\right)\) | \(e\left(\frac{203}{286}\right)\) | \(e\left(\frac{356}{1001}\right)\) | \(e\left(\frac{1871}{2002}\right)\) | \(e\left(\frac{67}{143}\right)\) |
| \(\chi_{6009}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{286}\right)\) | \(e\left(\frac{6}{143}\right)\) | \(e\left(\frac{1591}{2002}\right)\) | \(e\left(\frac{249}{1001}\right)\) | \(e\left(\frac{161}{286}\right)\) | \(e\left(\frac{316}{1001}\right)\) | \(e\left(\frac{269}{286}\right)\) | \(e\left(\frac{290}{1001}\right)\) | \(e\left(\frac{1541}{2002}\right)\) | \(e\left(\frac{12}{143}\right)\) |
| \(\chi_{6009}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{286}\right)\) | \(e\left(\frac{58}{143}\right)\) | \(e\left(\frac{603}{2002}\right)\) | \(e\left(\frac{548}{1001}\right)\) | \(e\left(\frac{31}{286}\right)\) | \(e\left(\frac{4}{1001}\right)\) | \(e\left(\frac{217}{286}\right)\) | \(e\left(\frac{992}{1001}\right)\) | \(e\left(\frac{501}{2002}\right)\) | \(e\left(\frac{116}{143}\right)\) |
| \(\chi_{6009}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{286}\right)\) | \(e\left(\frac{96}{143}\right)\) | \(e\left(\frac{145}{2002}\right)\) | \(e\left(\frac{409}{1001}\right)\) | \(e\left(\frac{145}{286}\right)\) | \(e\left(\frac{909}{1001}\right)\) | \(e\left(\frac{157}{286}\right)\) | \(e\left(\frac{207}{1001}\right)\) | \(e\left(\frac{489}{2002}\right)\) | \(e\left(\frac{49}{143}\right)\) |
| \(\chi_{6009}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{286}\right)\) | \(e\left(\frac{85}{143}\right)\) | \(e\left(\frac{1399}{2002}\right)\) | \(e\left(\frac{453}{1001}\right)\) | \(e\left(\frac{255}{286}\right)\) | \(e\left(\frac{997}{1001}\right)\) | \(e\left(\frac{69}{286}\right)\) | \(e\left(\frac{9}{1001}\right)\) | \(e\left(\frac{1501}{2002}\right)\) | \(e\left(\frac{27}{143}\right)\) |
| \(\chi_{6009}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{286}\right)\) | \(e\left(\frac{45}{143}\right)\) | \(e\left(\frac{1279}{2002}\right)\) | \(e\left(\frac{80}{1001}\right)\) | \(e\left(\frac{135}{286}\right)\) | \(e\left(\frac{797}{1001}\right)\) | \(e\left(\frac{87}{286}\right)\) | \(e\left(\frac{459}{1001}\right)\) | \(e\left(\frac{475}{2002}\right)\) | \(e\left(\frac{90}{143}\right)\) |
| \(\chi_{6009}(140,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{286}\right)\) | \(e\left(\frac{27}{143}\right)\) | \(e\left(\frac{1797}{2002}\right)\) | \(e\left(\frac{906}{1001}\right)\) | \(e\left(\frac{81}{286}\right)\) | \(e\left(\frac{993}{1001}\right)\) | \(e\left(\frac{281}{286}\right)\) | \(e\left(\frac{18}{1001}\right)\) | \(e\left(\frac{2001}{2002}\right)\) | \(e\left(\frac{54}{143}\right)\) |
| \(\chi_{6009}(161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{286}\right)\) | \(e\left(\frac{131}{143}\right)\) | \(e\left(\frac{1537}{2002}\right)\) | \(e\left(\frac{932}{1001}\right)\) | \(e\left(\frac{107}{286}\right)\) | \(e\left(\frac{226}{1001}\right)\) | \(e\left(\frac{177}{286}\right)\) | \(e\left(\frac{993}{1001}\right)\) | \(e\left(\frac{779}{2002}\right)\) | \(e\left(\frac{119}{143}\right)\) |
| \(\chi_{6009}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{286}\right)\) | \(e\left(\frac{42}{143}\right)\) | \(e\left(\frac{1985}{2002}\right)\) | \(e\left(\frac{456}{1001}\right)\) | \(e\left(\frac{269}{286}\right)\) | \(e\left(\frac{639}{1001}\right)\) | \(e\left(\frac{167}{286}\right)\) | \(e\left(\frac{314}{1001}\right)\) | \(e\left(\frac{205}{2002}\right)\) | \(e\left(\frac{84}{143}\right)\) |
| \(\chi_{6009}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{286}\right)\) | \(e\left(\frac{113}{143}\right)\) | \(e\left(\frac{625}{2002}\right)\) | \(e\left(\frac{900}{1001}\right)\) | \(e\left(\frac{53}{286}\right)\) | \(e\left(\frac{708}{1001}\right)\) | \(e\left(\frac{85}{286}\right)\) | \(e\left(\frac{409}{1001}\right)\) | \(e\left(\frac{589}{2002}\right)\) | \(e\left(\frac{83}{143}\right)\) |
| \(\chi_{6009}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{286}\right)\) | \(e\left(\frac{92}{143}\right)\) | \(e\left(\frac{1563}{2002}\right)\) | \(e\left(\frac{529}{1001}\right)\) | \(e\left(\frac{133}{286}\right)\) | \(e\left(\frac{603}{1001}\right)\) | \(e\left(\frac{73}{286}\right)\) | \(e\left(\frac{395}{1001}\right)\) | \(e\left(\frac{701}{2002}\right)\) | \(e\left(\frac{41}{143}\right)\) |
| \(\chi_{6009}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{286}\right)\) | \(e\left(\frac{53}{143}\right)\) | \(e\left(\frac{445}{2002}\right)\) | \(e\left(\frac{841}{1001}\right)\) | \(e\left(\frac{159}{286}\right)\) | \(e\left(\frac{408}{1001}\right)\) | \(e\left(\frac{255}{286}\right)\) | \(e\left(\frac{83}{1001}\right)\) | \(e\left(\frac{51}{2002}\right)\) | \(e\left(\frac{106}{143}\right)\) |
| \(\chi_{6009}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{286}\right)\) | \(e\left(\frac{64}{143}\right)\) | \(e\left(\frac{1479}{2002}\right)\) | \(e\left(\frac{368}{1001}\right)\) | \(e\left(\frac{49}{286}\right)\) | \(e\left(\frac{463}{1001}\right)\) | \(e\left(\frac{57}{286}\right)\) | \(e\left(\frac{710}{1001}\right)\) | \(e\left(\frac{183}{2002}\right)\) | \(e\left(\frac{128}{143}\right)\) |
| \(\chi_{6009}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{286}\right)\) | \(e\left(\frac{67}{143}\right)\) | \(e\left(\frac{1059}{2002}\right)\) | \(e\left(\frac{564}{1001}\right)\) | \(e\left(\frac{201}{286}\right)\) | \(e\left(\frac{764}{1001}\right)\) | \(e\left(\frac{263}{286}\right)\) | \(e\left(\frac{283}{1001}\right)\) | \(e\left(\frac{1597}{2002}\right)\) | \(e\left(\frac{134}{143}\right)\) |
| \(\chi_{6009}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{286}\right)\) | \(e\left(\frac{43}{143}\right)\) | \(e\left(\frac{701}{2002}\right)\) | \(e\left(\frac{569}{1001}\right)\) | \(e\left(\frac{129}{286}\right)\) | \(e\left(\frac{501}{1001}\right)\) | \(e\left(\frac{45}{286}\right)\) | \(e\left(\frac{124}{1001}\right)\) | \(e\left(\frac{1439}{2002}\right)\) | \(e\left(\frac{86}{143}\right)\) |
| \(\chi_{6009}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{286}\right)\) | \(e\left(\frac{87}{143}\right)\) | \(e\left(\frac{547}{2002}\right)\) | \(e\left(\frac{107}{1001}\right)\) | \(e\left(\frac{261}{286}\right)\) | \(e\left(\frac{578}{1001}\right)\) | \(e\left(\frac{111}{286}\right)\) | \(e\left(\frac{201}{1001}\right)\) | \(e\left(\frac{823}{2002}\right)\) | \(e\left(\frac{31}{143}\right)\) |
| \(\chi_{6009}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{286}\right)\) | \(e\left(\frac{92}{143}\right)\) | \(e\left(\frac{1849}{2002}\right)\) | \(e\left(\frac{100}{1001}\right)\) | \(e\left(\frac{133}{286}\right)\) | \(e\left(\frac{746}{1001}\right)\) | \(e\left(\frac{73}{286}\right)\) | \(e\left(\frac{824}{1001}\right)\) | \(e\left(\frac{1845}{2002}\right)\) | \(e\left(\frac{41}{143}\right)\) |
| \(\chi_{6009}(215,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{286}\right)\) | \(e\left(\frac{85}{143}\right)\) | \(e\left(\frac{255}{2002}\right)\) | \(e\left(\frac{167}{1001}\right)\) | \(e\left(\frac{255}{286}\right)\) | \(e\left(\frac{425}{1001}\right)\) | \(e\left(\frac{69}{286}\right)\) | \(e\left(\frac{295}{1001}\right)\) | \(e\left(\frac{929}{2002}\right)\) | \(e\left(\frac{27}{143}\right)\) |
| \(\chi_{6009}(218,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{286}\right)\) | \(e\left(\frac{10}{143}\right)\) | \(e\left(\frac{745}{2002}\right)\) | \(e\left(\frac{272}{1001}\right)\) | \(e\left(\frac{173}{286}\right)\) | \(e\left(\frac{908}{1001}\right)\) | \(e\left(\frac{67}{286}\right)\) | \(e\left(\frac{960}{1001}\right)\) | \(e\left(\frac{1615}{2002}\right)\) | \(e\left(\frac{20}{143}\right)\) |
| \(\chi_{6009}(224,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{286}\right)\) | \(e\left(\frac{29}{143}\right)\) | \(e\left(\frac{1803}{2002}\right)\) | \(e\left(\frac{274}{1001}\right)\) | \(e\left(\frac{87}{286}\right)\) | \(e\left(\frac{2}{1001}\right)\) | \(e\left(\frac{37}{286}\right)\) | \(e\left(\frac{496}{1001}\right)\) | \(e\left(\frac{751}{2002}\right)\) | \(e\left(\frac{58}{143}\right)\) |
| \(\chi_{6009}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{286}\right)\) | \(e\left(\frac{58}{143}\right)\) | \(e\left(\frac{317}{2002}\right)\) | \(e\left(\frac{977}{1001}\right)\) | \(e\left(\frac{31}{286}\right)\) | \(e\left(\frac{862}{1001}\right)\) | \(e\left(\frac{217}{286}\right)\) | \(e\left(\frac{563}{1001}\right)\) | \(e\left(\frac{1359}{2002}\right)\) | \(e\left(\frac{116}{143}\right)\) |
| \(\chi_{6009}(248,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{286}\right)\) | \(e\left(\frac{34}{143}\right)\) | \(e\left(\frac{531}{2002}\right)\) | \(e\left(\frac{124}{1001}\right)\) | \(e\left(\frac{245}{286}\right)\) | \(e\left(\frac{885}{1001}\right)\) | \(e\left(\frac{285}{286}\right)\) | \(e\left(\frac{261}{1001}\right)\) | \(e\left(\frac{1487}{2002}\right)\) | \(e\left(\frac{68}{143}\right)\) |
| \(\chi_{6009}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{286}\right)\) | \(e\left(\frac{141}{143}\right)\) | \(e\left(\frac{995}{2002}\right)\) | \(e\left(\frac{632}{1001}\right)\) | \(e\left(\frac{137}{286}\right)\) | \(e\left(\frac{991}{1001}\right)\) | \(e\left(\frac{101}{286}\right)\) | \(e\left(\frac{523}{1001}\right)\) | \(e\left(\frac{249}{2002}\right)\) | \(e\left(\frac{139}{143}\right)\) |
| \(\chi_{6009}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{286}\right)\) | \(e\left(\frac{95}{143}\right)\) | \(e\left(\frac{571}{2002}\right)\) | \(e\left(\frac{582}{1001}\right)\) | \(e\left(\frac{285}{286}\right)\) | \(e\left(\frac{618}{1001}\right)\) | \(e\left(\frac{279}{286}\right)\) | \(e\left(\frac{111}{1001}\right)\) | \(e\left(\frac{1829}{2002}\right)\) | \(e\left(\frac{47}{143}\right)\) |
| \(\chi_{6009}(266,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{237}{286}\right)\) | \(e\left(\frac{94}{143}\right)\) | \(e\left(\frac{997}{2002}\right)\) | \(e\left(\frac{755}{1001}\right)\) | \(e\left(\frac{139}{286}\right)\) | \(e\left(\frac{327}{1001}\right)\) | \(e\left(\frac{115}{286}\right)\) | \(e\left(\frac{15}{1001}\right)\) | \(e\left(\frac{1167}{2002}\right)\) | \(e\left(\frac{45}{143}\right)\) |