from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,243))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(243\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 81.h | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{243}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) |
| \(\chi_{243}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{243}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{243}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) |
| \(\chi_{243}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) |
| \(\chi_{243}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) |
| \(\chi_{243}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) |
| \(\chi_{243}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) |
| \(\chi_{243}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{243}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) |
| \(\chi_{243}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{243}(152,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) |
| \(\chi_{243}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{243}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{243}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) |
| \(\chi_{243}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) |
| \(\chi_{243}(224,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) |
| \(\chi_{243}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) |