from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,23]))
pari: [g,chi] = znchar(Mod(356,6069))
Basic properties
| Modulus: | \(6069\) | |
| Conductor: | \(6069\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6069.bq
\(\chi_{6069}(356,\cdot)\) \(\chi_{6069}(713,\cdot)\) \(\chi_{6069}(1070,\cdot)\) \(\chi_{6069}(1427,\cdot)\) \(\chi_{6069}(1784,\cdot)\) \(\chi_{6069}(2141,\cdot)\) \(\chi_{6069}(2498,\cdot)\) \(\chi_{6069}(2855,\cdot)\) \(\chi_{6069}(3212,\cdot)\) \(\chi_{6069}(3569,\cdot)\) \(\chi_{6069}(3926,\cdot)\) \(\chi_{6069}(4283,\cdot)\) \(\chi_{6069}(4640,\cdot)\) \(\chi_{6069}(4997,\cdot)\) \(\chi_{6069}(5354,\cdot)\) \(\chi_{6069}(5711,\cdot)\)
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
\((2024,4336,3760)\) → \((-1,-1,e\left(\frac{23}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(356, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) |
sage: chi.jacobi_sum(n)