from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([10,6,1]))
pari: [g,chi] = znchar(Mod(32,6039))
Basic properties
| Modulus: | \(6039\) | |
| Conductor: | \(6039\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6039.ec
\(\chi_{6039}(32,\cdot)\) \(\chi_{6039}(1748,\cdot)\) \(\chi_{6039}(2408,\cdot)\) \(\chi_{6039}(4718,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | Number field defined by a degree 12 polynomial |
Values on generators
\((1343,5491,5248)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6039 }(32, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)