from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([28]))
pari: [g,chi] = znchar(Mod(27,229))
Basic properties
Modulus: | \(229\) | |
Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | 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.g
\(\chi_{229}(16,\cdot)\) \(\chi_{229}(17,\cdot)\) \(\chi_{229}(27,\cdot)\) \(\chi_{229}(42,\cdot)\) \(\chi_{229}(43,\cdot)\) \(\chi_{229}(44,\cdot)\) \(\chi_{229}(53,\cdot)\) \(\chi_{229}(57,\cdot)\) \(\chi_{229}(60,\cdot)\) \(\chi_{229}(61,\cdot)\) \(\chi_{229}(104,\cdot)\) \(\chi_{229}(121,\cdot)\) \(\chi_{229}(161,\cdot)\) \(\chi_{229}(165,\cdot)\) \(\chi_{229}(203,\cdot)\) \(\chi_{229}(214,\cdot)\) \(\chi_{229}(218,\cdot)\) \(\chi_{229}(225,\cdot)\)
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 19 polynomial |
Values on generators
\(6\) → \(e\left(\frac{14}{19}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 229 }(27, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) |
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)