from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1033, base_ring=CyclotomicField(172))
M = H._module
chi = DirichletCharacter(H, M([63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,1033))
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: | \(172\) | 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: | $\Q(\zeta_{172})$ |
| Fixed field: | Number field defined by a degree 172 polynomial (not computed) |
First 31 of 84 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1033}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{63}{172}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{119}{172}\right)\) | \(e\left(\frac{163}{172}\right)\) |
| \(\chi_{1033}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{59}{172}\right)\) | \(e\left(\frac{23}{172}\right)\) |
| \(\chi_{1033}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{7}{172}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{147}{172}\right)\) | \(e\left(\frac{171}{172}\right)\) |
| \(\chi_{1033}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{135}{172}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{83}{172}\right)\) | \(e\left(\frac{79}{172}\right)\) |
| \(\chi_{1033}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{73}{172}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{157}{172}\right)\) | \(e\left(\frac{137}{172}\right)\) |
| \(\chi_{1033}(50,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{61}{172}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{77}{172}\right)\) | \(e\left(\frac{65}{172}\right)\) |
| \(\chi_{1033}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{21}{172}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{97}{172}\right)\) | \(e\left(\frac{169}{172}\right)\) |
| \(\chi_{1033}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{47}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{127}{172}\right)\) | \(e\left(\frac{67}{172}\right)\) |
| \(\chi_{1033}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{111}{172}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{95}{172}\right)\) | \(e\left(\frac{107}{172}\right)\) |
| \(\chi_{1033}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{81}{172}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{13}{172}\right)\) |
| \(\chi_{1033}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{155}{172}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{159}{172}\right)\) | \(e\left(\frac{27}{172}\right)\) |
| \(\chi_{1033}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{1}{172}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{21}{172}\right)\) | \(e\left(\frac{49}{172}\right)\) |
| \(\chi_{1033}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{111}{172}\right)\) | \(e\left(\frac{87}{172}\right)\) |
| \(\chi_{1033}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{45}{172}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{141}{172}\right)\) |
| \(\chi_{1033}(166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{89}{172}\right)\) |
| \(\chi_{1033}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{23}{172}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{139}{172}\right)\) | \(e\left(\frac{95}{172}\right)\) |
| \(\chi_{1033}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{89}{172}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{149}{172}\right)\) | \(e\left(\frac{61}{172}\right)\) |
| \(\chi_{1033}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{17}{172}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{13}{172}\right)\) | \(e\left(\frac{145}{172}\right)\) |
| \(\chi_{1033}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{105}{172}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{141}{172}\right)\) | \(e\left(\frac{157}{172}\right)\) |
| \(\chi_{1033}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{5}{172}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{105}{172}\right)\) | \(e\left(\frac{73}{172}\right)\) |
| \(\chi_{1033}(228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{117}{172}\right)\) | \(e\left(\frac{101}{172}\right)\) |
| \(\chi_{1033}(242,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{77}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{69}{172}\right)\) | \(e\left(\frac{161}{172}\right)\) |
| \(\chi_{1033}(252,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{137}{172}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{125}{172}\right)\) | \(e\left(\frac{5}{172}\right)\) |
| \(\chi_{1033}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{39}{172}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{131}{172}\right)\) | \(e\left(\frac{19}{172}\right)\) |
| \(\chi_{1033}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{53}{172}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{81}{172}\right)\) | \(e\left(\frac{17}{172}\right)\) |
| \(\chi_{1033}(266,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{101}{172}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{57}{172}\right)\) | \(e\left(\frac{133}{172}\right)\) |
| \(\chi_{1033}(272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{145}{172}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{121}{172}\right)\) | \(e\left(\frac{53}{172}\right)\) |
| \(\chi_{1033}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{55}{172}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{123}{172}\right)\) | \(e\left(\frac{115}{172}\right)\) |
| \(\chi_{1033}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{95}{172}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{11}{172}\right)\) |
| \(\chi_{1033}(290,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{29}{172}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{93}{172}\right)\) | \(e\left(\frac{45}{172}\right)\) |
| \(\chi_{1033}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{37}{172}\right)\) |