from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4760, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,0,3,10,3]))
pari: [g,chi] = znchar(Mod(47,4760))
Basic properties
| Modulus: | \(4760\) | |
| Conductor: | \(2380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2380}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4760.fp
\(\chi_{4760}(47,\cdot)\) \(\chi_{4760}(1823,\cdot)\) \(\chi_{4760}(3183,\cdot)\) \(\chi_{4760}(3447,\cdot)\)
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
\((1191,2381,2857,1361,3641)\) → \((-1,1,i,e\left(\frac{5}{6}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 4760 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)