from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1033, base_ring=CyclotomicField(344))
M = H._module
chi = DirichletCharacter(H, M([5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,1033))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1033\) | |
| Conductor: | \(1033\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(344\) | 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: | $\Q(\zeta_{344})$ |
| Fixed field: | Number field defined by a degree 344 polynomial (not computed) |
First 31 of 168 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1033}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{35}{172}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{5}{344}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{165}{172}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{105}{344}\right)\) | \(e\left(\frac{73}{344}\right)\) |
| \(\chi_{1033}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{243}{344}\right)\) | \(e\left(\frac{3}{172}\right)\) | \(e\left(\frac{107}{172}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{287}{344}\right)\) | \(e\left(\frac{39}{344}\right)\) |
| \(\chi_{1033}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{147}{172}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{193}{344}\right)\) | \(e\left(\frac{13}{172}\right)\) | \(e\left(\frac{5}{172}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{269}{344}\right)\) | \(e\left(\frac{341}{344}\right)\) |
| \(\chi_{1033}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{55}{172}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{229}{344}\right)\) | \(e\left(\frac{109}{172}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{337}{344}\right)\) | \(e\left(\frac{41}{344}\right)\) |
| \(\chi_{1033}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{93}{172}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{87}{344}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{119}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{107}{344}\right)\) | \(e\left(\frac{307}{344}\right)\) |
| \(\chi_{1033}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{83}{172}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{233}{344}\right)\) | \(e\left(\frac{5}{172}\right)\) | \(e\left(\frac{121}{172}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{77}{344}\right)\) | \(e\left(\frac{237}{344}\right)\) |
| \(\chi_{1033}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{1}{172}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{123}{344}\right)\) | \(e\left(\frac{27}{172}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{175}{344}\right)\) | \(e\left(\frac{7}{344}\right)\) |
| \(\chi_{1033}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{117}{172}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{115}{344}\right)\) | \(e\left(\frac{63}{172}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{7}{344}\right)\) | \(e\left(\frac{303}{344}\right)\) |
| \(\chi_{1033}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{143}{172}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{217}{344}\right)\) | \(e\left(\frac{77}{172}\right)\) | \(e\left(\frac{109}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{85}{344}\right)\) | \(e\left(\frac{141}{344}\right)\) |
| \(\chi_{1033}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{149}{344}\right)\) | \(e\left(\frac{125}{172}\right)\) | \(e\left(\frac{101}{172}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{33}{344}\right)\) | \(e\left(\frac{249}{344}\right)\) |
| \(\chi_{1033}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{285}{344}\right)\) | \(e\left(\frac{29}{172}\right)\) | \(e\left(\frac{117}{172}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{137}{344}\right)\) | \(e\left(\frac{33}{344}\right)\) |
| \(\chi_{1033}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{145}{172}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{119}{344}\right)\) | \(e\left(\frac{131}{172}\right)\) | \(e\left(\frac{143}{172}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{91}{344}\right)\) | \(e\left(\frac{155}{344}\right)\) |
| \(\chi_{1033}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{57}{172}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{131}{344}\right)\) | \(e\left(\frac{163}{172}\right)\) | \(e\left(\frac{23}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{343}{344}\right)\) | \(e\left(\frac{55}{344}\right)\) |
| \(\chi_{1033}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{69}{172}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{59}{344}\right)\) | \(e\left(\frac{143}{172}\right)\) | \(e\left(\frac{55}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{207}{344}\right)\) | \(e\left(\frac{311}{344}\right)\) |
| \(\chi_{1033}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{41}{172}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{55}{344}\right)\) | \(e\left(\frac{75}{172}\right)\) | \(e\left(\frac{95}{172}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{123}{344}\right)\) | \(e\left(\frac{115}{344}\right)\) |
| \(\chi_{1033}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{17}{172}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{27}{344}\right)\) | \(e\left(\frac{115}{172}\right)\) | \(e\left(\frac{31}{172}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{223}{344}\right)\) | \(e\left(\frac{119}{344}\right)\) |
| \(\chi_{1033}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{49}{172}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{179}{344}\right)\) | \(e\left(\frac{119}{172}\right)\) | \(e\left(\frac{59}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{319}{344}\right)\) | \(e\left(\frac{343}{344}\right)\) |
| \(\chi_{1033}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{5}{172}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{271}{344}\right)\) | \(e\left(\frac{135}{172}\right)\) | \(e\left(\frac{171}{172}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{187}{344}\right)\) | \(e\left(\frac{35}{344}\right)\) |
| \(\chi_{1033}(120,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{3}{172}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{25}{344}\right)\) | \(e\left(\frac{81}{172}\right)\) | \(e\left(\frac{137}{172}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{181}{344}\right)\) | \(e\left(\frac{21}{344}\right)\) |
| \(\chi_{1033}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{7}{172}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{1}{344}\right)\) | \(e\left(\frac{17}{172}\right)\) | \(e\left(\frac{33}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{21}{344}\right)\) | \(e\left(\frac{221}{344}\right)\) |
| \(\chi_{1033}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{67}{172}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{329}{344}\right)\) | \(e\left(\frac{89}{172}\right)\) | \(e\left(\frac{21}{172}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{29}{344}\right)\) | \(e\left(\frac{125}{344}\right)\) |
| \(\chi_{1033}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{87}{172}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{209}{344}\right)\) | \(e\left(\frac{113}{172}\right)\) | \(e\left(\frac{17}{172}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{261}{344}\right)\) | \(e\left(\frac{93}{344}\right)\) |
| \(\chi_{1033}(140,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{135}{172}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{265}{344}\right)\) | \(e\left(\frac{33}{172}\right)\) | \(e\left(\frac{145}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{61}{344}\right)\) | \(e\left(\frac{85}{344}\right)\) |
| \(\chi_{1033}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{101}{172}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{39}{344}\right)\) | \(e\left(\frac{147}{172}\right)\) | \(e\left(\frac{83}{172}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{131}{344}\right)\) | \(e\left(\frac{19}{344}\right)\) |
| \(\chi_{1033}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{73}{172}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{35}{344}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{123}{172}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{47}{344}\right)\) | \(e\left(\frac{167}{344}\right)\) |
| \(\chi_{1033}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{19}{172}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{101}{344}\right)\) | \(e\left(\frac{169}{172}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{57}{344}\right)\) | \(e\left(\frac{305}{344}\right)\) |
| \(\chi_{1033}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{195}{344}\right)\) | \(e\left(\frac{47}{172}\right)\) | \(e\left(\frac{71}{172}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{311}{344}\right)\) | \(e\left(\frac{95}{344}\right)\) |
| \(\chi_{1033}(160,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{121}{172}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{263}{344}\right)\) | \(e\left(\frac{171}{172}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{19}{344}\right)\) | \(e\left(\frac{331}{344}\right)\) |
| \(\chi_{1033}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{95}{172}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{333}{344}\right)\) | \(e\left(\frac{157}{172}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{113}{344}\right)\) | \(e\left(\frac{321}{344}\right)\) |
| \(\chi_{1033}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{123}{172}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{337}{344}\right)\) | \(e\left(\frac{53}{172}\right)\) | \(e\left(\frac{113}{172}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{197}{344}\right)\) | \(e\left(\frac{173}{344}\right)\) |
| \(\chi_{1033}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{47}{172}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{277}{344}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{25}{172}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{313}{344}\right)\) | \(e\left(\frac{329}{344}\right)\) |