from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([16940,2420,3528]))
pari: [g,chi] = znchar(Mod(718,586971))
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(586971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12705\) | 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 586971.rh
\(\chi_{586971}(25,\cdot)\) \(\chi_{586971}(58,\cdot)\) \(\chi_{586971}(247,\cdot)\) \(\chi_{586971}(466,\cdot)\) \(\chi_{586971}(499,\cdot)\) \(\chi_{586971}(592,\cdot)\) \(\chi_{586971}(625,\cdot)\) \(\chi_{586971}(718,\cdot)\) \(\chi_{586971}(751,\cdot)\) \(\chi_{586971}(907,\cdot)\) \(\chi_{586971}(940,\cdot)\) \(\chi_{586971}(1159,\cdot)\) \(\chi_{586971}(1192,\cdot)\) \(\chi_{586971}(1285,\cdot)\) \(\chi_{586971}(1318,\cdot)\) \(\chi_{586971}(1411,\cdot)\) \(\chi_{586971}(1444,\cdot)\)
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: | $\Q(\zeta_{12705})$ |
| Fixed field: | Number field defined by a degree 12705 polynomial (not computed) |
Values on generators
\((130439,179686,73207)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{21}\right),e\left(\frac{84}{605}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(718, a) \) | \(1\) | \(1\) | \(e\left(\frac{1193}{4235}\right)\) | \(e\left(\frac{2386}{4235}\right)\) | \(e\left(\frac{8161}{12705}\right)\) | \(e\left(\frac{3579}{4235}\right)\) | \(e\left(\frac{2348}{2541}\right)\) | \(e\left(\frac{2879}{12705}\right)\) | \(e\left(\frac{537}{4235}\right)\) | \(e\left(\frac{5806}{12705}\right)\) | \(e\left(\frac{566}{1815}\right)\) | \(e\left(\frac{2614}{12705}\right)\) |
sage: chi.jacobi_sum(n)