from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(86190, base_ring=CyclotomicField(208))
M = H._module
chi = DirichletCharacter(H, M([104,0,192,169]))
chi.galois_orbit()
[g,chi] = znchar(Mod(131,86190))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(86190\) | |
| Conductor: | \(8619\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(208\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 8619.ev | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{86190}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{145}{208}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{187}{208}\right)\) | \(e\left(\frac{25}{104}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{86190}(521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{208}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{99}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{35}{104}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{86190}(911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{83}{208}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{71}{104}\right)\) | \(e\left(\frac{17}{52}\right)\) |
| \(\chi_{86190}(2081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{113}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{43}{208}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) |
| \(\chi_{86190}(2471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{121}{208}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{86190}(2861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{208}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{17}{104}\right)\) | \(e\left(\frac{7}{52}\right)\) |
| \(\chi_{86190}(3641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{173}{208}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{37}{104}\right)\) | \(e\left(\frac{3}{52}\right)\) |
| \(\chi_{86190}(5981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{157}{208}\right)\) | \(e\left(\frac{53}{208}\right)\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{43}{52}\right)\) |
| \(\chi_{86190}(7151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{199}{208}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{145}{208}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) |
| \(\chi_{86190}(7541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{208}\right)\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{99}{208}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{103}{208}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{86190}(8711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{97}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{59}{208}\right)\) | \(e\left(\frac{199}{208}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{29}{52}\right)\) |
| \(\chi_{86190}(9101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{63}{208}\right)\) | \(e\left(\frac{75}{208}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{1}{52}\right)\) |
| \(\chi_{86190}(9491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{63}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{197}{208}\right)\) | \(e\left(\frac{185}{208}\right)\) | \(e\left(\frac{81}{208}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{57}{104}\right)\) | \(e\left(\frac{51}{52}\right)\) |
| \(\chi_{86190}(10271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{3}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{49}{208}\right)\) | \(e\left(\frac{197}{208}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{135}{208}\right)\) | \(e\left(\frac{77}{104}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{86190}(12611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{208}\right)\) | \(e\left(\frac{187}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{85}{104}\right)\) | \(e\left(\frac{35}{52}\right)\) |
| \(\chi_{86190}(13391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{208}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{89}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{31}{52}\right)\) |
| \(\chi_{86190}(13781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{149}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{43}{208}\right)\) | \(e\left(\frac{49}{208}\right)\) | \(e\left(\frac{11}{104}\right)\) | \(e\left(\frac{29}{52}\right)\) |
| \(\chi_{86190}(14171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{208}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{115}{208}\right)\) | \(e\left(\frac{127}{208}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{47}{104}\right)\) | \(e\left(\frac{1}{52}\right)\) |
| \(\chi_{86190}(15341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{81}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{75}{208}\right)\) | \(e\left(\frac{119}{208}\right)\) | \(e\left(\frac{15}{208}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{103}{104}\right)\) | \(e\left(\frac{21}{52}\right)\) |
| \(\chi_{86190}(15731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{208}\right)\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{99}{208}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{35}{104}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{86190}(16121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{208}\right)\) | \(e\left(\frac{105}{208}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{35}{208}\right)\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{43}{52}\right)\) |
| \(\chi_{86190}(19241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{89}{208}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{21}{104}\right)\) | \(e\left(\frac{27}{52}\right)\) |
| \(\chi_{86190}(20021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{208}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{45}{208}\right)\) | \(e\left(\frac{113}{208}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{41}{104}\right)\) | \(e\left(\frac{23}{52}\right)\) |
| \(\chi_{86190}(20411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{208}\right)\) | \(e\left(\frac{133}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{67}{208}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{51}{104}\right)\) | \(e\left(\frac{21}{52}\right)\) |
| \(\chi_{86190}(20801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{151}{208}\right)\) | \(e\left(\frac{85}{208}\right)\) | \(e\left(\frac{87}{104}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{86190}(22361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{208}\right)\) | \(e\left(\frac{61}{208}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{41}{208}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) |
| \(\chi_{86190}(22751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{31}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{129}{208}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{33}{104}\right)\) | \(e\left(\frac{35}{52}\right)\) |
| \(\chi_{86190}(23531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{81}{208}\right)\) | \(e\left(\frac{37}{208}\right)\) | \(e\left(\frac{141}{208}\right)\) | \(e\left(\frac{151}{208}\right)\) | \(e\left(\frac{53}{104}\right)\) | \(e\left(\frac{31}{52}\right)\) |
| \(\chi_{86190}(25871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{105}{208}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{111}{208}\right)\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{19}{52}\right)\) |
| \(\chi_{86190}(26651,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{199}{208}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{61}{208}\right)\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{81}{104}\right)\) | \(e\left(\frac{15}{52}\right)\) |
| \(\chi_{86190}(27431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{197}{208}\right)\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) |