from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,22]))
pari: [g,chi] = znchar(Mod(13,784))
Basic properties
| Modulus: | \(784\) | |
| Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 784.bi
\(\chi_{784}(13,\cdot)\) \(\chi_{784}(69,\cdot)\) \(\chi_{784}(125,\cdot)\) \(\chi_{784}(181,\cdot)\) \(\chi_{784}(237,\cdot)\) \(\chi_{784}(349,\cdot)\) \(\chi_{784}(405,\cdot)\) \(\chi_{784}(461,\cdot)\) \(\chi_{784}(517,\cdot)\) \(\chi_{784}(573,\cdot)\) \(\chi_{784}(629,\cdot)\) \(\chi_{784}(741,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.0.271776353216347717810469630450516372938858574109997048774397001728.1 |
Values on generators
\((687,197,689)\) → \((1,-i,e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 784 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-i\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) |
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)