from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([9]))
pari: [g,chi] = znchar(Mod(8,257))
Basic properties
Modulus: | \(257\) | |
Conductor: | \(257\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | 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 257.e
\(\chi_{257}(2,\cdot)\) \(\chi_{257}(8,\cdot)\) \(\chi_{257}(32,\cdot)\) \(\chi_{257}(128,\cdot)\) \(\chi_{257}(129,\cdot)\) \(\chi_{257}(225,\cdot)\) \(\chi_{257}(249,\cdot)\) \(\chi_{257}(255,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
Fixed field: | Number field defined by a degree 16 polynomial |
Values on generators
\(3\) → \(e\left(\frac{9}{16}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 257 }(8, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) |
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)