from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3879, base_ring=CyclotomicField(86))
M = H._module
chi = DirichletCharacter(H, M([43,46]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,3879))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3879\) | |
| Conductor: | \(1293\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(86\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1293.k | 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: | $\Q(\zeta_{43})$ |
| Fixed field: | Number field defined by a degree 86 polynomial |
First 31 of 42 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3879}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) |
| \(\chi_{3879}(440,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) |
| \(\chi_{3879}(449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) |
| \(\chi_{3879}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) |
| \(\chi_{3879}(467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) |
| \(\chi_{3879}(485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) |
| \(\chi_{3879}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) |
| \(\chi_{3879}(512,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) |
| \(\chi_{3879}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) |
| \(\chi_{3879}(575,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) |
| \(\chi_{3879}(593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) |
| \(\chi_{3879}(647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) |
| \(\chi_{3879}(674,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) |
| \(\chi_{3879}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) |
| \(\chi_{3879}(755,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) |
| \(\chi_{3879}(917,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) |
| \(\chi_{3879}(926,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) |
| \(\chi_{3879}(1007,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) |
| \(\chi_{3879}(1079,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) |
| \(\chi_{3879}(1160,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) |
| \(\chi_{3879}(1295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) |
| \(\chi_{3879}(1403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) |
| \(\chi_{3879}(1421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) |
| \(\chi_{3879}(1583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) |
| \(\chi_{3879}(1727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) |
| \(\chi_{3879}(1736,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) |
| \(\chi_{3879}(1772,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) |
| \(\chi_{3879}(1889,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) |
| \(\chi_{3879}(1916,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) |
| \(\chi_{3879}(2159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) |
| \(\chi_{3879}(2375,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) |