from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(683, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([21]))
pari: [g,chi] = znchar(Mod(8,683))
Basic properties
Modulus: | \(683\) | |
Conductor: | \(683\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | 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 683.d
\(\chi_{683}(2,\cdot)\) \(\chi_{683}(8,\cdot)\) \(\chi_{683}(32,\cdot)\) \(\chi_{683}(128,\cdot)\) \(\chi_{683}(342,\cdot)\) \(\chi_{683}(427,\cdot)\) \(\chi_{683}(512,\cdot)\) \(\chi_{683}(619,\cdot)\) \(\chi_{683}(667,\cdot)\) \(\chi_{683}(679,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\(5\) → \(e\left(\frac{21}{22}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 683 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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)