from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([32,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(94,5239))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5239\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 169.i | 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: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 39 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5239}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) |
| \(\chi_{5239}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) |
| \(\chi_{5239}(497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{5239}(776,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{5239}(900,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) |
| \(\chi_{5239}(1179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{5239}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{5239}(1582,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{5239}(1706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{5239}(1985,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) |
| \(\chi_{5239}(2109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) |
| \(\chi_{5239}(2791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) |
| \(\chi_{5239}(2915,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{5239}(3194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) |
| \(\chi_{5239}(3318,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) |
| \(\chi_{5239}(3597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) |
| \(\chi_{5239}(3721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{5239}(4000,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) |
| \(\chi_{5239}(4124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) |
| \(\chi_{5239}(4403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) |
| \(\chi_{5239}(4527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{5239}(4806,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) |
| \(\chi_{5239}(4930,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) |
| \(\chi_{5239}(5209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) |