from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,12]))
pari: [g,chi] = znchar(Mod(4,387))
Basic properties
| Modulus: | \(387\) | |
| Conductor: | \(387\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | 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 387.ba
\(\chi_{387}(4,\cdot)\) \(\chi_{387}(16,\cdot)\) \(\chi_{387}(97,\cdot)\) \(\chi_{387}(121,\cdot)\) \(\chi_{387}(133,\cdot)\) \(\chi_{387}(193,\cdot)\) \(\chi_{387}(250,\cdot)\) \(\chi_{387}(256,\cdot)\) \(\chi_{387}(274,\cdot)\) \(\chi_{387}(322,\cdot)\) \(\chi_{387}(355,\cdot)\) \(\chi_{387}(385,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((173,46)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 387 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)