from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, base_ring=CyclotomicField(86))
M = H._module
chi = DirichletCharacter(H, M([0,43,0,44]))
chi.galois_orbit()
[g,chi] = znchar(Mod(169,8304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8304\) | |
| Conductor: | \(1384\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(86\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1384.s | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8304}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) |
| \(\chi_{8304}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) |
| \(\chi_{8304}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) |
| \(\chi_{8304}(1081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) |
| \(\chi_{8304}(1177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) |
| \(\chi_{8304}(1225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) |
| \(\chi_{8304}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) |
| \(\chi_{8304}(1465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) |
| \(\chi_{8304}(1609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) |
| \(\chi_{8304}(1657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) |
| \(\chi_{8304}(1705,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) |
| \(\chi_{8304}(1753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) |
| \(\chi_{8304}(1849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) |
| \(\chi_{8304}(2041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) |
| \(\chi_{8304}(2185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) |
| \(\chi_{8304}(2473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) |
| \(\chi_{8304}(2617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{10}{43}\right)\) |
| \(\chi_{8304}(2713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) |
| \(\chi_{8304}(3001,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) |
| \(\chi_{8304}(3577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) |
| \(\chi_{8304}(3769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) |
| \(\chi_{8304}(4489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) |
| \(\chi_{8304}(4633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) |
| \(\chi_{8304}(4681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) |
| \(\chi_{8304}(4777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) |
| \(\chi_{8304}(4873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) |
| \(\chi_{8304}(5113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) |
| \(\chi_{8304}(5593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) |
| \(\chi_{8304}(5833,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) |
| \(\chi_{8304}(5929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) |
| \(\chi_{8304}(5977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) |