from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,517))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(517\) | |
| Conductor: | \(517\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 46 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{517}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{517}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{517}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{517}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) |
| \(\chi_{517}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{517}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{517}(153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{517}(175,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{517}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{517}(230,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{517}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{517}(252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{517}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{517}(285,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{517}(296,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{517}(307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{517}(318,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{517}(384,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{517}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{517}(450,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{517}(472,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{517}(494,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) |