from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1352, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,46]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,1352))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1352\) | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1352}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(705,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(945,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(1017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(1121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(1153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1352}(1225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1352}(1257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |