from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([49,26]))
chi.galois_orbit()
[g,chi] = znchar(Mod(603,8003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(8003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(52\) | 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: | $\Q(\zeta_{52})$ |
| Fixed field: | Number field defined by a degree 52 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8003}(603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) |
| \(\chi_{8003}(754,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) |
| \(\chi_{8003}(1207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) |
| \(\chi_{8003}(1358,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) |
| \(\chi_{8003}(2566,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) |
| \(\chi_{8003}(2717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{8003}(3019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
| \(\chi_{8003}(3321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{8003}(3472,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{8003}(3623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{8003}(3925,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) |
| \(\chi_{8003}(4076,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{8003}(4378,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{8003}(4831,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) |
| \(\chi_{8003}(5133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) |
| \(\chi_{8003}(5586,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{8003}(5888,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{8003}(6039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) |
| \(\chi_{8003}(6341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{8003}(6492,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{8003}(6643,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{8003}(6945,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
| \(\chi_{8003}(7247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{8003}(7398,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) |