from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(927, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,22]))
pari: [g,chi] = znchar(Mod(179,927))
Basic properties
| Modulus: | \(927\) | |
| Conductor: | \(309\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{309}(179,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 927.x
\(\chi_{927}(8,\cdot)\) \(\chi_{927}(116,\cdot)\) \(\chi_{927}(179,\cdot)\) \(\chi_{927}(215,\cdot)\) \(\chi_{927}(278,\cdot)\) \(\chi_{927}(287,\cdot)\) \(\chi_{927}(323,\cdot)\) \(\chi_{927}(332,\cdot)\) \(\chi_{927}(476,\cdot)\) \(\chi_{927}(512,\cdot)\) \(\chi_{927}(755,\cdot)\) \(\chi_{927}(782,\cdot)\) \(\chi_{927}(800,\cdot)\) \(\chi_{927}(854,\cdot)\) \(\chi_{927}(890,\cdot)\) \(\chi_{927}(917,\cdot)\)
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: | 34.0.3325466068076643664357827857598159738994734276327509143073421552355865283.1 |
Values on generators
\((722,829)\) → \((-1,e\left(\frac{11}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 927 }(179, a) \) | \(-1\) | \(1\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) |
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)