from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(485, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([36,35]))
pari: [g,chi] = znchar(Mod(3,485))
Basic properties
Modulus: | \(485\) | |
Conductor: | \(485\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 485.bm
\(\chi_{485}(2,\cdot)\) \(\chi_{485}(3,\cdot)\) \(\chi_{485}(32,\cdot)\) \(\chi_{485}(48,\cdot)\) \(\chi_{485}(72,\cdot)\) \(\chi_{485}(108,\cdot)\) \(\chi_{485}(122,\cdot)\) \(\chi_{485}(128,\cdot)\) \(\chi_{485}(162,\cdot)\) \(\chi_{485}(163,\cdot)\) \(\chi_{485}(183,\cdot)\) \(\chi_{485}(192,\cdot)\) \(\chi_{485}(243,\cdot)\) \(\chi_{485}(247,\cdot)\) \(\chi_{485}(288,\cdot)\) \(\chi_{485}(432,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((292,296)\) → \((-i,e\left(\frac{35}{48}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 485 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) |
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)