from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(265, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([13,14]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,265))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(265\) | |
| Conductor: | \(265\) | 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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{265}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) |
| \(\chi_{265}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) |
| \(\chi_{265}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) |
| \(\chi_{265}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) |
| \(\chi_{265}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) |
| \(\chi_{265}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) |
| \(\chi_{265}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) |
| \(\chi_{265}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{33}{52}\right)\) |
| \(\chi_{265}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) |
| \(\chi_{265}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{17}{52}\right)\) |
| \(\chi_{265}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) |
| \(\chi_{265}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) |
| \(\chi_{265}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) |
| \(\chi_{265}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{265}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) |
| \(\chi_{265}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{265}(168,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) |
| \(\chi_{265}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) |
| \(\chi_{265}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) |
| \(\chi_{265}(202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{265}(218,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{29}{52}\right)\) |
| \(\chi_{265}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) |
| \(\chi_{265}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) |
| \(\chi_{265}(252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) |