from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(485, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,9]))
pari: [g,chi] = znchar(Mod(186,485))
Basic properties
Modulus: | \(485\) | |
Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{97}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 485.y
\(\chi_{485}(176,\cdot)\) \(\chi_{485}(186,\cdot)\) \(\chi_{485}(206,\cdot)\) \(\chi_{485}(221,\cdot)\) \(\chi_{485}(361,\cdot)\) \(\chi_{485}(376,\cdot)\) \(\chi_{485}(396,\cdot)\) \(\chi_{485}(406,\cdot)\)
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
\((292,296)\) → \((1,e\left(\frac{9}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 485 }(186, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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)