from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(779, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,24]))
pari: [g,chi] = znchar(Mod(406,779))
Basic properties
Modulus: | \(779\) | |
Conductor: | \(779\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(15\) | 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 779.t
\(\chi_{779}(182,\cdot)\) \(\chi_{779}(201,\cdot)\) \(\chi_{779}(387,\cdot)\) \(\chi_{779}(406,\cdot)\) \(\chi_{779}(467,\cdot)\) \(\chi_{779}(543,\cdot)\) \(\chi_{779}(672,\cdot)\) \(\chi_{779}(748,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((534,457)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 779 }(406, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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)