from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(208))
M = H._module
chi = DirichletCharacter(H, M([0,28,117]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,8619))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8619\) | |
| Conductor: | \(2873\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(208\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2873.cj | 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: | $\Q(\zeta_{208})$ |
| Fixed field: | Number field defined by a degree 208 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8619}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{5}{208}\right)\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{3}{104}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{167}{208}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{104}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{141}{208}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{111}{208}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{185}{208}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{49}{208}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{97}{208}\right)\) | \(e\left(\frac{15}{208}\right)\) | \(e\left(\frac{79}{104}\right)\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{173}{208}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{83}{104}\right)\) | \(e\left(\frac{159}{208}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{104}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{113}{208}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{104}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{5}{208}\right)\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{115}{208}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{104}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{145}{208}\right)\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{37}{208}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(694,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{104}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{37}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{43}{104}\right)\) | \(e\left(\frac{135}{208}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{104}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{47}{104}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{17}{208}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{129}{208}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{15}{104}\right)\) | \(e\left(\frac{139}{208}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{105}{208}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(1006,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{104}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{19}{104}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{133}{208}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{104}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{81}{208}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{139}{208}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(1048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{139}{208}\right)\) | \(e\left(\frac{133}{208}\right)\) | \(e\left(\frac{21}{104}\right)\) | \(e\left(\frac{153}{208}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(1357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{75}{208}\right)\) | \(e\left(\frac{83}{104}\right)\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{61}{208}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(1438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{104}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{89}{208}\right)\) | \(e\left(\frac{25}{104}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(1474,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{135}{208}\right)\) | \(e\left(\frac{87}{104}\right)\) | \(e\left(\frac{99}{208}\right)\) | \(e\left(\frac{163}{208}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(1552,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{104}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{55}{104}\right)\) | \(e\left(\frac{59}{208}\right)\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(1669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(1672,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{115}{208}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{69}{104}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{97}{208}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(1711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{53}{208}\right)\) | \(e\left(\frac{85}{104}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{121}{208}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(1945,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{104}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{111}{208}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{25}{104}\right)\) | \(e\left(\frac{197}{208}\right)\) | \(e\left(\frac{5}{208}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(2020,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{104}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{19}{104}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{87}{208}\right)\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(2101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{173}{208}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(2137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{104}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{67}{208}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(2215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{104}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{95}{104}\right)\) | \(e\left(\frac{187}{208}\right)\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{8619}(2332,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{104}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{61}{208}\right)\) | \(e\left(\frac{3}{208}\right)\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{127}{208}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{8619}(2335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{104}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{83}{208}\right)\) | \(e\left(\frac{45}{208}\right)\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(2374,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{104}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{75}{208}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{105}{208}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{3}{208}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{8619}(2608,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{113}{208}\right)\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{103}{208}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{8619}(2683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{133}{208}\right)\) | \(e\left(\frac{27}{208}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{103}{208}\right)\) | \(e\left(\frac{199}{208}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{8}\right)\) |