from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(959, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,22]))
pari: [g,chi] = znchar(Mod(818,959))
Basic properties
| Modulus: | \(959\) | |
| Conductor: | \(959\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | 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 959.t
\(\chi_{959}(34,\cdot)\) \(\chi_{959}(153,\cdot)\) \(\chi_{959}(209,\cdot)\) \(\chi_{959}(461,\cdot)\) \(\chi_{959}(608,\cdot)\) \(\chi_{959}(622,\cdot)\) \(\chi_{959}(636,\cdot)\) \(\chi_{959}(671,\cdot)\) \(\chi_{959}(741,\cdot)\) \(\chi_{959}(804,\cdot)\) \(\chi_{959}(818,\cdot)\) \(\chi_{959}(860,\cdot)\) \(\chi_{959}(881,\cdot)\) \(\chi_{959}(895,\cdot)\) \(\chi_{959}(937,\cdot)\) \(\chi_{959}(944,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\((549,414)\) → \((-1,e\left(\frac{11}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 959 }(818, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(-1\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) |
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)