from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,21]))
pari: [g,chi] = znchar(Mod(526,867))
Basic properties
| Modulus: | \(867\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{289}(237,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 867.l
\(\chi_{867}(16,\cdot)\) \(\chi_{867}(67,\cdot)\) \(\chi_{867}(118,\cdot)\) \(\chi_{867}(169,\cdot)\) \(\chi_{867}(220,\cdot)\) \(\chi_{867}(271,\cdot)\) \(\chi_{867}(322,\cdot)\) \(\chi_{867}(373,\cdot)\) \(\chi_{867}(424,\cdot)\) \(\chi_{867}(475,\cdot)\) \(\chi_{867}(526,\cdot)\) \(\chi_{867}(628,\cdot)\) \(\chi_{867}(679,\cdot)\) \(\chi_{867}(730,\cdot)\) \(\chi_{867}(781,\cdot)\) \(\chi_{867}(832,\cdot)\)
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
\((290,292)\) → \((1,e\left(\frac{21}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 867 }(526, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) |
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)