from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1096, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,15]))
pari: [g,chi] = znchar(Mod(475,1096))
Basic properties
| Modulus: | \(1096\) | |
| Conductor: | \(1096\) | 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 1096.u
\(\chi_{1096}(99,\cdot)\) \(\chi_{1096}(155,\cdot)\) \(\chi_{1096}(323,\cdot)\) \(\chi_{1096}(339,\cdot)\) \(\chi_{1096}(355,\cdot)\) \(\chi_{1096}(395,\cdot)\) \(\chi_{1096}(475,\cdot)\) \(\chi_{1096}(563,\cdot)\) \(\chi_{1096}(611,\cdot)\) \(\chi_{1096}(635,\cdot)\) \(\chi_{1096}(651,\cdot)\) \(\chi_{1096}(699,\cdot)\) \(\chi_{1096}(707,\cdot)\) \(\chi_{1096}(763,\cdot)\) \(\chi_{1096}(899,\cdot)\) \(\chi_{1096}(963,\cdot)\)
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
\((823,549,825)\) → \((-1,-1,e\left(\frac{15}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1096 }(475, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) |
sage: chi.jacobi_sum(n)