from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6162, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,17]))
chi.galois_orbit()
[g,chi] = znchar(Mod(545,6162))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(6162\) | |
Conductor: | \(3081\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 3081.cj | 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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 26 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{6162}(545,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(-1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) |
\(\chi_{6162}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(-1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) |
\(\chi_{6162}(1325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(-1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) |
\(\chi_{6162}(1559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(-1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) |
\(\chi_{6162}(1637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(-1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) |
\(\chi_{6162}(2885,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(-1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) |
\(\chi_{6162}(3509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(-1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) |
\(\chi_{6162}(3977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(-1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) |
\(\chi_{6162}(4757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(-1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) |
\(\chi_{6162}(4913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(-1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) |
\(\chi_{6162}(4991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(-1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) |
\(\chi_{6162}(5147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(-1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) |