from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(597, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,43]))
chi.galois_orbit()
[g,chi] = znchar(Mod(55,597))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(597\) | |
| Conductor: | \(199\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 199.j | 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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{597}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) |
| \(\chi_{597}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) |
| \(\chi_{597}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) |
| \(\chi_{597}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) |
| \(\chi_{597}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) |
| \(\chi_{597}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) |
| \(\chi_{597}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) |
| \(\chi_{597}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) |
| \(\chi_{597}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) |
| \(\chi_{597}(334,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) |
| \(\chi_{597}(346,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) |
| \(\chi_{597}(358,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) |
| \(\chi_{597}(370,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) |
| \(\chi_{597}(373,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
| \(\chi_{597}(415,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) |
| \(\chi_{597}(457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) |
| \(\chi_{597}(481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) |
| \(\chi_{597}(499,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) |
| \(\chi_{597}(589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) |
| \(\chi_{597}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) |