from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(431, base_ring=CyclotomicField(86))
M = H._module
chi = DirichletCharacter(H, M([44]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,431))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(431\) | |
| Conductor: | \(431\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(43\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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 43 polynomial |
First 31 of 42 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{431}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) |
| \(\chi_{431}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) |
| \(\chi_{431}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) |
| \(\chi_{431}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) |
| \(\chi_{431}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) |
| \(\chi_{431}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) |
| \(\chi_{431}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) |
| \(\chi_{431}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) |
| \(\chi_{431}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) |
| \(\chi_{431}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) |
| \(\chi_{431}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) |
| \(\chi_{431}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) |
| \(\chi_{431}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) |
| \(\chi_{431}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) |
| \(\chi_{431}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) |
| \(\chi_{431}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) |
| \(\chi_{431}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) |
| \(\chi_{431}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) |
| \(\chi_{431}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) |
| \(\chi_{431}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) |
| \(\chi_{431}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) |
| \(\chi_{431}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{28}{43}\right)\) |
| \(\chi_{431}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) |
| \(\chi_{431}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) |
| \(\chi_{431}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) |
| \(\chi_{431}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) |
| \(\chi_{431}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) |
| \(\chi_{431}(165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) |
| \(\chi_{431}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{15}{43}\right)\) |
| \(\chi_{431}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) |
| \(\chi_{431}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) |