from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,9,11]))
pari: [g,chi] = znchar(Mod(21,5185))
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1037}(21,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5185.dd
\(\chi_{5185}(21,\cdot)\) \(\chi_{5185}(1016,\cdot)\) \(\chi_{5185}(2716,\cdot)\) \(\chi_{5185}(3506,\cdot)\)
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
\((3112,4576,2381)\) → \((1,-i,e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 5185 }(21, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)