from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([19,8,10]))
pari: [g,chi] = znchar(Mod(588,5225))
Basic properties
Modulus: | \(5225\) | |
Conductor: | \(5225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5225.ds
\(\chi_{5225}(588,\cdot)\) \(\chi_{5225}(702,\cdot)\) \(\chi_{5225}(1158,\cdot)\) \(\chi_{5225}(1367,\cdot)\) \(\chi_{5225}(2887,\cdot)\) \(\chi_{5225}(3172,\cdot)\) \(\chi_{5225}(4673,\cdot)\) \(\chi_{5225}(5053,\cdot)\)
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
\((2927,2851,4676)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{2}{5}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 5225 }(588, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)