from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8670, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,17,32]))
pari: [g,chi] = znchar(Mod(409,8670))
Basic properties
| Modulus: | \(8670\) | |
| Conductor: | \(1445\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1445}(409,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8670.bp
\(\chi_{8670}(409,\cdot)\) \(\chi_{8670}(919,\cdot)\) \(\chi_{8670}(1429,\cdot)\) \(\chi_{8670}(1939,\cdot)\) \(\chi_{8670}(2449,\cdot)\) \(\chi_{8670}(2959,\cdot)\) \(\chi_{8670}(3979,\cdot)\) \(\chi_{8670}(4489,\cdot)\) \(\chi_{8670}(4999,\cdot)\) \(\chi_{8670}(5509,\cdot)\) \(\chi_{8670}(6019,\cdot)\) \(\chi_{8670}(6529,\cdot)\) \(\chi_{8670}(7039,\cdot)\) \(\chi_{8670}(7549,\cdot)\) \(\chi_{8670}(8059,\cdot)\) \(\chi_{8670}(8569,\cdot)\)
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
\((2891,6937,6361)\) → \((1,-1,e\left(\frac{16}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8670 }(409, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) |
sage: chi.jacobi_sum(n)