from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6171, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,6,5]))
pari: [g,chi] = znchar(Mod(965,6171))
Basic properties
Modulus: | \(6171\) | |
Conductor: | \(561\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{561}(404,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6171.bg
\(\chi_{6171}(965,\cdot)\) \(\chi_{6171}(1322,\cdot)\) \(\chi_{6171}(1976,\cdot)\) \(\chi_{6171}(2393,\cdot)\) \(\chi_{6171}(3506,\cdot)\) \(\chi_{6171}(3863,\cdot)\) \(\chi_{6171}(4934,\cdot)\) \(\chi_{6171}(5606,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((4115,970,2179)\) → \((-1,e\left(\frac{3}{10}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(19\) |
\( \chi_{ 6171 }(965, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)