from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([0,19360,23982]))
pari: [g,chi] = znchar(Mod(37,586971))
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(65219\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12705\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{65219}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 586971.rj
\(\chi_{586971}(37,\cdot)\) \(\chi_{586971}(163,\cdot)\) \(\chi_{586971}(235,\cdot)\) \(\chi_{586971}(289,\cdot)\) \(\chi_{586971}(478,\cdot)\) \(\chi_{586971}(676,\cdot)\) \(\chi_{586971}(730,\cdot)\) \(\chi_{586971}(982,\cdot)\) \(\chi_{586971}(1054,\cdot)\) \(\chi_{586971}(1171,\cdot)\) \(\chi_{586971}(1180,\cdot)\) \(\chi_{586971}(1369,\cdot)\) \(\chi_{586971}(1423,\cdot)\)
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
\((130439,179686,73207)\) → \((1,e\left(\frac{16}{21}\right),e\left(\frac{571}{605}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{9571}{12705}\right)\) | \(e\left(\frac{6437}{12705}\right)\) | \(e\left(\frac{9589}{12705}\right)\) | \(e\left(\frac{1101}{4235}\right)\) | \(e\left(\frac{1291}{2541}\right)\) | \(e\left(\frac{2747}{4235}\right)\) | \(e\left(\frac{169}{12705}\right)\) | \(e\left(\frac{1424}{12705}\right)\) | \(e\left(\frac{664}{1815}\right)\) | \(e\left(\frac{1107}{4235}\right)\) |
sage: chi.jacobi_sum(n)