from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(547, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([25]))
pari: [g,chi] = znchar(Mod(107,547))
Basic properties
Modulus: | \(547\) | |
Conductor: | \(547\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | 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 547.i
\(\chi_{547}(28,\cdot)\) \(\chi_{547}(30,\cdot)\) \(\chi_{547}(38,\cdot)\) \(\chi_{547}(72,\cdot)\) \(\chi_{547}(107,\cdot)\) \(\chi_{547}(172,\cdot)\) \(\chi_{547}(194,\cdot)\) \(\chi_{547}(197,\cdot)\) \(\chi_{547}(254,\cdot)\) \(\chi_{547}(286,\cdot)\) \(\chi_{547}(310,\cdot)\) \(\chi_{547}(501,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\(2\) → \(e\left(\frac{25}{26}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 547 }(107, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(-1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
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)