from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1352, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,0,56]))
chi.galois_orbit()
[g,chi] = znchar(Mod(55,1352))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1352\) | |
| Conductor: | \(676\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(78\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 676.t | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 78 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1352}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(471,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(575,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(711,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(1023,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(1095,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(1127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(1199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(1231,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1352}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1352}(1335,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{6}\right)\) |