from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(35,304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(304\) | |
Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\Q(\zeta_{36})\) |
Fixed field: | 36.0.52733281945045886724167383478270850720626086921526306402773390818541568.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{304}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{304}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{304}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{304}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{304}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{304}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{304}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{304}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{304}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{304}(275,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{304}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{304}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |