from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4760, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,0,6,4,21]))
pari: [g,chi] = znchar(Mod(87,4760))
Basic properties
| Modulus: | \(4760\) | |
| Conductor: | \(2380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2380}(87,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4760.jj
\(\chi_{4760}(87,\cdot)\) \(\chi_{4760}(383,\cdot)\) \(\chi_{4760}(943,\cdot)\) \(\chi_{4760}(1447,\cdot)\) \(\chi_{4760}(2327,\cdot)\) \(\chi_{4760}(3687,\cdot)\) \(\chi_{4760}(3783,\cdot)\) \(\chi_{4760}(4343,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((1191,2381,2857,1361,3641)\) → \((-1,1,i,e\left(\frac{1}{6}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 4760 }(87, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)