from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([4235,9075,3129]))
pari: [g,chi] = znchar(Mod(776,586971))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 586971.sc
\(\chi_{586971}(41,\cdot)\) \(\chi_{586971}(83,\cdot)\) \(\chi_{586971}(167,\cdot)\) \(\chi_{586971}(272,\cdot)\) \(\chi_{586971}(398,\cdot)\) \(\chi_{586971}(545,\cdot)\) \(\chi_{586971}(776,\cdot)\) \(\chi_{586971}(860,\cdot)\) \(\chi_{586971}(986,\cdot)\) \(\chi_{586971}(1091,\cdot)\) \(\chi_{586971}(1217,\cdot)\) \(\chi_{586971}(1238,\cdot)\) \(\chi_{586971}(1427,\cdot)\)
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
\((130439,179686,73207)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{14}\right),e\left(\frac{149}{1210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 586971 }(776, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7312}{12705}\right)\) | \(e\left(\frac{1919}{12705}\right)\) | \(e\left(\frac{10778}{12705}\right)\) | \(e\left(\frac{3077}{4235}\right)\) | \(e\left(\frac{359}{847}\right)\) | \(e\left(\frac{137}{12705}\right)\) | \(e\left(\frac{3838}{12705}\right)\) | \(e\left(\frac{67}{8470}\right)\) | \(e\left(\frac{381}{605}\right)\) | \(e\left(\frac{12697}{12705}\right)\) |
sage: chi.jacobi_sum(n)