from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([4235,15730,21]))
pari: [g,chi] = znchar(Mod(2,586971))
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(586971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25410\) | 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.sk
\(\chi_{586971}(2,\cdot)\) \(\chi_{586971}(95,\cdot)\) \(\chi_{586971}(347,\cdot)\) \(\chi_{586971}(380,\cdot)\) \(\chi_{586971}(536,\cdot)\) \(\chi_{586971}(662,\cdot)\) \(\chi_{586971}(695,\cdot)\) \(\chi_{586971}(788,\cdot)\) \(\chi_{586971}(821,\cdot)\) \(\chi_{586971}(1040,\cdot)\) \(\chi_{586971}(1073,\cdot)\) \(\chi_{586971}(1229,\cdot)\) \(\chi_{586971}(1262,\cdot)\) \(\chi_{586971}(1355,\cdot)\) \(\chi_{586971}(1388,\cdot)\) \(\chi_{586971}(1481,\cdot)\) \(\chi_{586971}(1514,\cdot)\)
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 25410 polynomial (not computed) |
Values on generators
\((130439,179686,73207)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{21}\right),e\left(\frac{1}{1210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(2, a) \) | \(1\) | \(1\) | \(e\left(\frac{3338}{12705}\right)\) | \(e\left(\frac{6676}{12705}\right)\) | \(e\left(\frac{7943}{8470}\right)\) | \(e\left(\frac{3338}{4235}\right)\) | \(e\left(\frac{1019}{5082}\right)\) | \(e\left(\frac{19171}{25410}\right)\) | \(e\left(\frac{647}{12705}\right)\) | \(e\left(\frac{1367}{12705}\right)\) | \(e\left(\frac{2009}{3630}\right)\) | \(e\left(\frac{11771}{25410}\right)\) |
sage: chi.jacobi_sum(n)