from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,21,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(86,6003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(6003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{6003}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) |
\(\chi_{6003}(434,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) |
\(\chi_{6003}(608,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) |
\(\chi_{6003}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) |
\(\chi_{6003}(1217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) |
\(\chi_{6003}(1391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) |
\(\chi_{6003}(1652,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) |
\(\chi_{6003}(1739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) |
\(\chi_{6003}(2261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) |
\(\chi_{6003}(2435,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) |
\(\chi_{6003}(2522,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) |
\(\chi_{6003}(2696,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) |
\(\chi_{6003}(3218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) |
\(\chi_{6003}(3740,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) |
\(\chi_{6003}(4088,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) |
\(\chi_{6003}(4262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) |
\(\chi_{6003}(4523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) |
\(\chi_{6003}(4610,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) |
\(\chi_{6003}(5393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) |
\(\chi_{6003}(5654,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) |