from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, base_ring=CyclotomicField(86))
M = H._module
chi = DirichletCharacter(H, M([43,0,43,81]))
chi.galois_orbit()
[g,chi] = znchar(Mod(383,8304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(8304\) | |
Conductor: | \(2076\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(86\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 2076.n | 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: | $\Q(\zeta_{43})$ |
Fixed field: | Number field defined by a degree 86 polynomial |
First 31 of 42 characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{8304}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{49}{86}\right)\) |
\(\chi_{8304}(575,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{53}{86}\right)\) |
\(\chi_{8304}(1151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{71}{86}\right)\) |
\(\chi_{8304}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{23}{86}\right)\) |
\(\chi_{8304}(1535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{63}{86}\right)\) |
\(\chi_{8304}(1679,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{69}{86}\right)\) |
\(\chi_{8304}(1967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{5}{86}\right)\) |
\(\chi_{8304}(2111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{83}{86}\right)\) |
\(\chi_{8304}(2303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{83}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{11}{86}\right)\) |
\(\chi_{8304}(2399,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{31}{86}\right)\) |
\(\chi_{8304}(2447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{65}{86}\right)\) |
\(\chi_{8304}(2495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{41}{86}\right)\) | \(e\left(\frac{81}{86}\right)\) |
\(\chi_{8304}(2543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{67}{86}\right)\) |
\(\chi_{8304}(2687,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{79}{86}\right)\) | \(e\left(\frac{47}{86}\right)\) |
\(\chi_{8304}(2783,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{55}{86}\right)\) |
\(\chi_{8304}(2927,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{75}{86}\right)\) | \(e\left(\frac{37}{86}\right)\) |
\(\chi_{8304}(2975,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{33}{86}\right)\) |
\(\chi_{8304}(3071,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{63}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{27}{86}\right)\) |
\(\chi_{8304}(3311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{25}{86}\right)\) |
\(\chi_{8304}(3839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{13}{86}\right)\) |
\(\chi_{8304}(3983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{59}{86}\right)\) |
\(\chi_{8304}(4319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{21}{86}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{9}{86}\right)\) |
\(\chi_{8304}(4415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{49}{86}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{21}{86}\right)\) |
\(\chi_{8304}(4511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{51}{86}\right)\) |
\(\chi_{8304}(4655,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{75}{86}\right)\) |
\(\chi_{8304}(5231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{29}{86}\right)\) |
\(\chi_{8304}(5279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{3}{86}\right)\) |
\(\chi_{8304}(5327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{85}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{33}{86}\right)\) | \(e\left(\frac{61}{86}\right)\) |
\(\chi_{8304}(5567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{85}{86}\right)\) |
\(\chi_{8304}(5903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{73}{86}\right)\) |
\(\chi_{8304}(6095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{35}{86}\right)\) |