from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([0,50,42]))
chi.galois_orbit()
[g,chi] = znchar(Mod(64,1617))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1617\) | |
| Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(35\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 539.v | 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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 35 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1617}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{1617}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{1617}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{1617}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{1617}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{1617}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{1617}(526,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{35}\right)\) |
| \(\chi_{1617}(610,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{1617}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{1617}(652,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{35}\right)\) |
| \(\chi_{1617}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{1617}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{1617}(862,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{1617}(988,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{1617}(1072,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{1617}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{1617}(1114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{1617}(1219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{6}{35}\right)\) |
| \(\chi_{1617}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{1617}(1345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{35}\right)\) |
| \(\chi_{1617}(1450,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{1617}(1534,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{35}\right)\) |
| \(\chi_{1617}(1555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{1617}(1576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{35}\right)\) |