from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(8470))
M = H._module
chi = DirichletCharacter(H, M([4235,7260,21]))
pari: [g,chi] = znchar(Mod(8,586971))
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(195657\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8470\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{195657}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 586971.rd
\(\chi_{586971}(8,\cdot)\) \(\chi_{586971}(134,\cdot)\) \(\chi_{586971}(260,\cdot)\) \(\chi_{586971}(512,\cdot)\) \(\chi_{586971}(701,\cdot)\) \(\chi_{586971}(827,\cdot)\) \(\chi_{586971}(953,\cdot)\) \(\chi_{586971}(1205,\cdot)\) \(\chi_{586971}(1394,\cdot)\)
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: | $\Q(\zeta_{4235})$ |
| Fixed field: | Number field defined by a degree 8470 polynomial (not computed) |
Values on generators
\((130439,179686,73207)\) → \((-1,e\left(\frac{6}{7}\right),e\left(\frac{3}{1210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(8, a) \) | \(1\) | \(1\) | \(e\left(\frac{3338}{4235}\right)\) | \(e\left(\frac{2441}{4235}\right)\) | \(e\left(\frac{6889}{8470}\right)\) | \(e\left(\frac{1544}{4235}\right)\) | \(e\left(\frac{1019}{1694}\right)\) | \(e\left(\frac{2231}{8470}\right)\) | \(e\left(\frac{647}{4235}\right)\) | \(e\left(\frac{1367}{4235}\right)\) | \(e\left(\frac{799}{1210}\right)\) | \(e\left(\frac{3301}{8470}\right)\) |
sage: chi.jacobi_sum(n)