from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1113, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,12]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1113))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1113\) | |
Conductor: | \(371\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 371.n | 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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 26 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1113}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{1113}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{1113}(307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
\(\chi_{1113}(328,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) |
\(\chi_{1113}(664,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{1113}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{1113}(811,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{1113}(895,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1113}(916,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
\(\chi_{1113}(937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) |
\(\chi_{1113}(1000,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{1113}(1084,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |