from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(259, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([12,10]))
pari: [g,chi] = znchar(Mod(144,259))
Basic properties
| Modulus: | \(259\) | |
| Conductor: | \(259\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | 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 259.y
\(\chi_{259}(9,\cdot)\) \(\chi_{259}(16,\cdot)\) \(\chi_{259}(81,\cdot)\) \(\chi_{259}(86,\cdot)\) \(\chi_{259}(144,\cdot)\) \(\chi_{259}(256,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.413239695274351729.1 |
Values on generators
\((38,113)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 259 }(144, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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)