from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2163, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,29]))
pari: [g,chi] = znchar(Mod(1175,2163))
Basic properties
| Modulus: | \(2163\) | |
| Conductor: | \(2163\) | 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 2163.bu
\(\chi_{2163}(125,\cdot)\) \(\chi_{2163}(209,\cdot)\) \(\chi_{2163}(230,\cdot)\) \(\chi_{2163}(398,\cdot)\) \(\chi_{2163}(713,\cdot)\) \(\chi_{2163}(1007,\cdot)\) \(\chi_{2163}(1175,\cdot)\) \(\chi_{2163}(1469,\cdot)\) \(\chi_{2163}(1511,\cdot)\) \(\chi_{2163}(1532,\cdot)\) \(\chi_{2163}(1658,\cdot)\) \(\chi_{2163}(1679,\cdot)\) \(\chi_{2163}(1721,\cdot)\) \(\chi_{2163}(1742,\cdot)\) \(\chi_{2163}(1994,\cdot)\) \(\chi_{2163}(2099,\cdot)\)
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
\((722,619,211)\) → \((-1,-1,e\left(\frac{29}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2163 }(1175, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) |
sage: chi.jacobi_sum(n)