from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4019, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([2]))
pari: [g,chi] = znchar(Mod(273,4019))
Basic properties
| Modulus: | \(4019\) | |
| Conductor: | \(4019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | 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 4019.c
\(\chi_{4019}(273,\cdot)\) \(\chi_{4019}(359,\cdot)\) \(\chi_{4019}(1428,\cdot)\) \(\chi_{4019}(1551,\cdot)\) \(\chi_{4019}(2187,\cdot)\) \(\chi_{4019}(2239,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.4214131050856513421881.1 |
Values on generators
\(2\) → \(e\left(\frac{1}{7}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4019 }(273, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)