from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,243))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(243\) | |
| Conductor: | \(243\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(81\) | 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: | $\Q(\zeta_{81})$ |
| Fixed field: | Number field defined by a degree 81 polynomial |
First 31 of 54 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{243}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) |
| \(\chi_{243}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{59}{81}\right)\) |
| \(\chi_{243}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) |
| \(\chi_{243}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{59}{81}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) |
| \(\chi_{243}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{38}{81}\right)\) | \(e\left(\frac{1}{81}\right)\) |
| \(\chi_{243}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) |
| \(\chi_{243}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) |
| \(\chi_{243}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{68}{81}\right)\) |
| \(\chi_{243}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) |
| \(\chi_{243}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{17}{81}\right)\) |
| \(\chi_{243}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{59}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) |
| \(\chi_{243}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{38}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) |
| \(\chi_{243}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{38}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) |
| \(\chi_{243}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) |
| \(\chi_{243}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) |
| \(\chi_{243}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) |
| \(\chi_{243}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{73}{81}\right)\) |
| \(\chi_{243}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{29}{81}\right)\) |
| \(\chi_{243}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{31}{81}\right)\) |
| \(\chi_{243}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) |
| \(\chi_{243}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{64}{81}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{43}{81}\right)\) |
| \(\chi_{243}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{35}{81}\right)\) |
| \(\chi_{243}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) |
| \(\chi_{243}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{38}{81}\right)\) |
| \(\chi_{243}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{67}{81}\right)\) |
| \(\chi_{243}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{59}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) |
| \(\chi_{243}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) |
| \(\chi_{243}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) |
| \(\chi_{243}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{64}{81}\right)\) |
| \(\chi_{243}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{64}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) |
| \(\chi_{243}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) |