from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([11]))
pari: [g,chi] = znchar(Mod(125,229))
Basic properties
Modulus: | \(229\) | |
Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | 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 229.h
\(\chi_{229}(4,\cdot)\) \(\chi_{229}(11,\cdot)\) \(\chi_{229}(15,\cdot)\) \(\chi_{229}(26,\cdot)\) \(\chi_{229}(64,\cdot)\) \(\chi_{229}(68,\cdot)\) \(\chi_{229}(108,\cdot)\) \(\chi_{229}(125,\cdot)\) \(\chi_{229}(168,\cdot)\) \(\chi_{229}(169,\cdot)\) \(\chi_{229}(172,\cdot)\) \(\chi_{229}(176,\cdot)\) \(\chi_{229}(185,\cdot)\) \(\chi_{229}(186,\cdot)\) \(\chi_{229}(187,\cdot)\) \(\chi_{229}(202,\cdot)\) \(\chi_{229}(212,\cdot)\) \(\chi_{229}(213,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\(6\) → \(e\left(\frac{11}{38}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 229 }(125, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) |
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)