from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([29]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,109))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(109\) | |
| Conductor: | \(109\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | 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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{109}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) |
| \(\chi_{109}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) |
| \(\chi_{109}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{19}{54}\right)\) |
| \(\chi_{109}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) |
| \(\chi_{109}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{41}{54}\right)\) |
| \(\chi_{109}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) |
| \(\chi_{109}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) |
| \(\chi_{109}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) |
| \(\chi_{109}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) |
| \(\chi_{109}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{43}{54}\right)\) |
| \(\chi_{109}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) |
| \(\chi_{109}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) |
| \(\chi_{109}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) |
| \(\chi_{109}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{47}{54}\right)\) |
| \(\chi_{109}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{23}{54}\right)\) |
| \(\chi_{109}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) |
| \(\chi_{109}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) |
| \(\chi_{109}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) |