from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([50]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,59))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(59\) | |
| Conductor: | \(59\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(29\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{29})$ |
| Fixed field: | Number field defined by a degree 29 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{59}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) |
| \(\chi_{59}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) |
| \(\chi_{59}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) |
| \(\chi_{59}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) |
| \(\chi_{59}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) |
| \(\chi_{59}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) |
| \(\chi_{59}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) |
| \(\chi_{59}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) |
| \(\chi_{59}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) |
| \(\chi_{59}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) |
| \(\chi_{59}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) |
| \(\chi_{59}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) |
| \(\chi_{59}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) |
| \(\chi_{59}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) |
| \(\chi_{59}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) |
| \(\chi_{59}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) |
| \(\chi_{59}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) |
| \(\chi_{59}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) |
| \(\chi_{59}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) |
| \(\chi_{59}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) |
| \(\chi_{59}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) |
| \(\chi_{59}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) |
| \(\chi_{59}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) |
| \(\chi_{59}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) |
| \(\chi_{59}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) |
| \(\chi_{59}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) |
| \(\chi_{59}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) |
| \(\chi_{59}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) |