from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(723, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,17]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,723))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(723\) | |
| Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 241.p | 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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{723}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(304,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(394,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(460,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(520,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(634,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{723}(658,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{723}(712,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |