from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(464))
M = H._module
chi = DirichletCharacter(H, M([29,400]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1003\) | |
| Conductor: | \(1003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(464\) | 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: | $\Q(\zeta_{464})$ |
| Fixed field: | Number field defined by a degree 464 polynomial (not computed) |
First 31 of 224 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1003}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{232}\right)\) | \(e\left(\frac{77}{464}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{225}{464}\right)\) | \(e\left(\frac{419}{464}\right)\) | \(e\left(\frac{95}{464}\right)\) | \(e\left(\frac{49}{232}\right)\) | \(e\left(\frac{77}{232}\right)\) | \(e\left(\frac{103}{464}\right)\) | \(e\left(\frac{459}{464}\right)\) |
| \(\chi_{1003}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{232}\right)\) | \(e\left(\frac{225}{464}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{85}{464}\right)\) | \(e\left(\frac{447}{464}\right)\) | \(e\left(\frac{139}{464}\right)\) | \(e\left(\frac{101}{232}\right)\) | \(e\left(\frac{225}{232}\right)\) | \(e\left(\frac{307}{464}\right)\) | \(e\left(\frac{359}{464}\right)\) |
| \(\chi_{1003}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{232}\right)\) | \(e\left(\frac{95}{464}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{139}{464}\right)\) | \(e\left(\frac{65}{464}\right)\) | \(e\left(\frac{69}{464}\right)\) | \(e\left(\frac{187}{232}\right)\) | \(e\left(\frac{95}{232}\right)\) | \(e\left(\frac{109}{464}\right)\) | \(e\left(\frac{265}{464}\right)\) |
| \(\chi_{1003}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{232}\right)\) | \(e\left(\frac{297}{464}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{205}{464}\right)\) | \(e\left(\frac{423}{464}\right)\) | \(e\left(\frac{35}{464}\right)\) | \(e\left(\frac{189}{232}\right)\) | \(e\left(\frac{65}{232}\right)\) | \(e\left(\frac{331}{464}\right)\) | \(e\left(\frac{47}{464}\right)\) |
| \(\chi_{1003}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{232}\right)\) | \(e\left(\frac{445}{464}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{65}{464}\right)\) | \(e\left(\frac{451}{464}\right)\) | \(e\left(\frac{79}{464}\right)\) | \(e\left(\frac{9}{232}\right)\) | \(e\left(\frac{213}{232}\right)\) | \(e\left(\frac{71}{464}\right)\) | \(e\left(\frac{411}{464}\right)\) |
| \(\chi_{1003}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{232}\right)\) | \(e\left(\frac{337}{464}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{117}{464}\right)\) | \(e\left(\frac{255}{464}\right)\) | \(e\left(\frac{235}{464}\right)\) | \(e\left(\frac{109}{232}\right)\) | \(e\left(\frac{105}{232}\right)\) | \(e\left(\frac{35}{464}\right)\) | \(e\left(\frac{183}{464}\right)\) |
| \(\chi_{1003}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{232}\right)\) | \(e\left(\frac{231}{464}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{211}{464}\right)\) | \(e\left(\frac{329}{464}\right)\) | \(e\left(\frac{285}{464}\right)\) | \(e\left(\frac{147}{232}\right)\) | \(e\left(\frac{231}{232}\right)\) | \(e\left(\frac{309}{464}\right)\) | \(e\left(\frac{449}{464}\right)\) |
| \(\chi_{1003}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{232}\right)\) | \(e\left(\frac{315}{464}\right)\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{119}{464}\right)\) | \(e\left(\frac{69}{464}\right)\) | \(e\left(\frac{9}{464}\right)\) | \(e\left(\frac{95}{232}\right)\) | \(e\left(\frac{83}{232}\right)\) | \(e\left(\frac{337}{464}\right)\) | \(e\left(\frac{317}{464}\right)\) |
| \(\chi_{1003}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{232}\right)\) | \(e\left(\frac{441}{464}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{445}{464}\right)\) | \(e\left(\frac{375}{464}\right)\) | \(e\left(\frac{291}{464}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{209}{232}\right)\) | \(e\left(\frac{379}{464}\right)\) | \(e\left(\frac{351}{464}\right)\) |
| \(\chi_{1003}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{232}\right)\) | \(e\left(\frac{351}{464}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{411}{464}\right)\) | \(e\left(\frac{289}{464}\right)\) | \(e\left(\frac{421}{464}\right)\) | \(e\left(\frac{139}{232}\right)\) | \(e\left(\frac{119}{232}\right)\) | \(e\left(\frac{349}{464}\right)\) | \(e\left(\frac{393}{464}\right)\) |
| \(\chi_{1003}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{232}\right)\) | \(e\left(\frac{379}{464}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{71}{464}\right)\) | \(e\left(\frac{357}{464}\right)\) | \(e\left(\frac{329}{464}\right)\) | \(e\left(\frac{199}{232}\right)\) | \(e\left(\frac{147}{232}\right)\) | \(e\left(\frac{49}{464}\right)\) | \(e\left(\frac{349}{464}\right)\) |
| \(\chi_{1003}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{232}\right)\) | \(e\left(\frac{281}{464}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{333}{464}\right)\) | \(e\left(\frac{119}{464}\right)\) | \(e\left(\frac{419}{464}\right)\) | \(e\left(\frac{221}{232}\right)\) | \(e\left(\frac{49}{232}\right)\) | \(e\left(\frac{171}{464}\right)\) | \(e\left(\frac{271}{464}\right)\) |
| \(\chi_{1003}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{232}\right)\) | \(e\left(\frac{53}{464}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{185}{464}\right)\) | \(e\left(\frac{427}{464}\right)\) | \(e\left(\frac{439}{464}\right)\) | \(e\left(\frac{97}{232}\right)\) | \(e\left(\frac{53}{232}\right)\) | \(e\left(\frac{95}{464}\right)\) | \(e\left(\frac{99}{464}\right)\) |
| \(\chi_{1003}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{232}\right)\) | \(e\left(\frac{371}{464}\right)\) | \(e\left(\frac{33}{116}\right)\) | \(e\left(\frac{367}{464}\right)\) | \(e\left(\frac{205}{464}\right)\) | \(e\left(\frac{289}{464}\right)\) | \(e\left(\frac{215}{232}\right)\) | \(e\left(\frac{139}{232}\right)\) | \(e\left(\frac{201}{464}\right)\) | \(e\left(\frac{229}{464}\right)\) |
| \(\chi_{1003}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{232}\right)\) | \(e\left(\frac{251}{464}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{167}{464}\right)\) | \(e\left(\frac{245}{464}\right)\) | \(e\left(\frac{153}{464}\right)\) | \(e\left(\frac{223}{232}\right)\) | \(e\left(\frac{19}{232}\right)\) | \(e\left(\frac{161}{464}\right)\) | \(e\left(\frac{285}{464}\right)\) |
| \(\chi_{1003}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{232}\right)\) | \(e\left(\frac{249}{464}\right)\) | \(e\left(\frac{95}{116}\right)\) | \(e\left(\frac{125}{464}\right)\) | \(e\left(\frac{439}{464}\right)\) | \(e\left(\frac{259}{464}\right)\) | \(e\left(\frac{53}{232}\right)\) | \(e\left(\frac{17}{232}\right)\) | \(e\left(\frac{315}{464}\right)\) | \(e\left(\frac{255}{464}\right)\) |
| \(\chi_{1003}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{232}\right)\) | \(e\left(\frac{413}{464}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{321}{464}\right)\) | \(e\left(\frac{307}{464}\right)\) | \(e\left(\frac{383}{464}\right)\) | \(e\left(\frac{73}{232}\right)\) | \(e\left(\frac{181}{232}\right)\) | \(e\left(\frac{215}{464}\right)\) | \(e\left(\frac{395}{464}\right)\) |
| \(\chi_{1003}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{232}\right)\) | \(e\left(\frac{99}{464}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{223}{464}\right)\) | \(e\left(\frac{141}{464}\right)\) | \(e\left(\frac{321}{464}\right)\) | \(e\left(\frac{63}{232}\right)\) | \(e\left(\frac{99}{232}\right)\) | \(e\left(\frac{265}{464}\right)\) | \(e\left(\frac{325}{464}\right)\) |
| \(\chi_{1003}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{232}\right)\) | \(e\left(\frac{63}{464}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{395}{464}\right)\) | \(e\left(\frac{385}{464}\right)\) | \(e\left(\frac{373}{464}\right)\) | \(e\left(\frac{19}{232}\right)\) | \(e\left(\frac{63}{232}\right)\) | \(e\left(\frac{253}{464}\right)\) | \(e\left(\frac{249}{464}\right)\) |
| \(\chi_{1003}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{232}\right)\) | \(e\left(\frac{439}{464}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{403}{464}\right)\) | \(e\left(\frac{105}{464}\right)\) | \(e\left(\frac{397}{464}\right)\) | \(e\left(\frac{195}{232}\right)\) | \(e\left(\frac{207}{232}\right)\) | \(e\left(\frac{69}{464}\right)\) | \(e\left(\frac{321}{464}\right)\) |
| \(\chi_{1003}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{232}\right)\) | \(e\left(\frac{155}{464}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{7}{464}\right)\) | \(e\left(\frac{277}{464}\right)\) | \(e\left(\frac{137}{464}\right)\) | \(e\left(\frac{183}{232}\right)\) | \(e\left(\frac{155}{232}\right)\) | \(e\left(\frac{129}{464}\right)\) | \(e\left(\frac{237}{464}\right)\) |
| \(\chi_{1003}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{232}\right)\) | \(e\left(\frac{201}{464}\right)\) | \(e\left(\frac{11}{116}\right)\) | \(e\left(\frac{45}{464}\right)\) | \(e\left(\frac{455}{464}\right)\) | \(e\left(\frac{19}{464}\right)\) | \(e\left(\frac{149}{232}\right)\) | \(e\left(\frac{201}{232}\right)\) | \(e\left(\frac{299}{464}\right)\) | \(e\left(\frac{463}{464}\right)\) |
| \(\chi_{1003}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{232}\right)\) | \(e\left(\frac{93}{464}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{97}{464}\right)\) | \(e\left(\frac{259}{464}\right)\) | \(e\left(\frac{175}{464}\right)\) | \(e\left(\frac{17}{232}\right)\) | \(e\left(\frac{93}{232}\right)\) | \(e\left(\frac{263}{464}\right)\) | \(e\left(\frac{235}{464}\right)\) |
| \(\chi_{1003}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{232}\right)\) | \(e\left(\frac{55}{464}\right)\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{227}{464}\right)\) | \(e\left(\frac{233}{464}\right)\) | \(e\left(\frac{333}{464}\right)\) | \(e\left(\frac{35}{232}\right)\) | \(e\left(\frac{55}{232}\right)\) | \(e\left(\frac{405}{464}\right)\) | \(e\left(\frac{129}{464}\right)\) |
| \(\chi_{1003}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{232}\right)\) | \(e\left(\frac{397}{464}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{449}{464}\right)\) | \(e\left(\frac{3}{464}\right)\) | \(e\left(\frac{303}{464}\right)\) | \(e\left(\frac{105}{232}\right)\) | \(e\left(\frac{165}{232}\right)\) | \(e\left(\frac{55}{464}\right)\) | \(e\left(\frac{155}{464}\right)\) |
| \(\chi_{1003}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{232}\right)\) | \(e\left(\frac{401}{464}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{69}{464}\right)\) | \(e\left(\frac{79}{464}\right)\) | \(e\left(\frac{91}{464}\right)\) | \(e\left(\frac{213}{232}\right)\) | \(e\left(\frac{169}{232}\right)\) | \(e\left(\frac{211}{464}\right)\) | \(e\left(\frac{215}{464}\right)\) |
| \(\chi_{1003}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{232}\right)\) | \(e\left(\frac{451}{464}\right)\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{191}{464}\right)\) | \(e\left(\frac{333}{464}\right)\) | \(e\left(\frac{225}{464}\right)\) | \(e\left(\frac{55}{232}\right)\) | \(e\left(\frac{219}{232}\right)\) | \(e\left(\frac{73}{464}\right)\) | \(e\left(\frac{37}{464}\right)\) |
| \(\chi_{1003}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{232}\right)\) | \(e\left(\frac{71}{464}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{99}{464}\right)\) | \(e\left(\frac{73}{464}\right)\) | \(e\left(\frac{413}{464}\right)\) | \(e\left(\frac{3}{232}\right)\) | \(e\left(\frac{71}{232}\right)\) | \(e\left(\frac{101}{464}\right)\) | \(e\left(\frac{369}{464}\right)\) |
| \(\chi_{1003}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{232}\right)\) | \(e\left(\frac{197}{464}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{425}{464}\right)\) | \(e\left(\frac{379}{464}\right)\) | \(e\left(\frac{231}{464}\right)\) | \(e\left(\frac{41}{232}\right)\) | \(e\left(\frac{197}{232}\right)\) | \(e\left(\frac{143}{464}\right)\) | \(e\left(\frac{403}{464}\right)\) |
| \(\chi_{1003}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{232}\right)\) | \(e\left(\frac{365}{464}\right)\) | \(e\left(\frac{95}{116}\right)\) | \(e\left(\frac{241}{464}\right)\) | \(e\left(\frac{323}{464}\right)\) | \(e\left(\frac{143}{464}\right)\) | \(e\left(\frac{169}{232}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{199}{464}\right)\) | \(e\left(\frac{139}{464}\right)\) |
| \(\chi_{1003}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{232}\right)\) | \(e\left(\frac{211}{464}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{255}{464}\right)\) | \(e\left(\frac{413}{464}\right)\) | \(e\left(\frac{417}{464}\right)\) | \(e\left(\frac{71}{232}\right)\) | \(e\left(\frac{211}{232}\right)\) | \(e\left(\frac{457}{464}\right)\) | \(e\left(\frac{149}{464}\right)\) |