from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(265, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,31]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,265))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(265\) | |
Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(52\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 53.f | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{265}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{4}{13}\right)\) |
\(\chi_{265}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) |
\(\chi_{265}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{3}{13}\right)\) |
\(\chi_{265}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{10}{13}\right)\) |
\(\chi_{265}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{6}{13}\right)\) |
\(\chi_{265}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{11}{13}\right)\) |
\(\chi_{265}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{5}{13}\right)\) |
\(\chi_{265}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{2}{13}\right)\) |
\(\chi_{265}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) |
\(\chi_{265}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) |
\(\chi_{265}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) |
\(\chi_{265}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) |
\(\chi_{265}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{2}{13}\right)\) |
\(\chi_{265}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{5}{13}\right)\) |
\(\chi_{265}(156,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{11}{13}\right)\) |
\(\chi_{265}(161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{6}{13}\right)\) |
\(\chi_{265}(171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{10}{13}\right)\) |
\(\chi_{265}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{3}{13}\right)\) |
\(\chi_{265}(186,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{7}{13}\right)\) |
\(\chi_{265}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{4}{13}\right)\) |
\(\chi_{265}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) |
\(\chi_{265}(231,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) |
\(\chi_{265}(246,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) |
\(\chi_{265}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) |