from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(327184, base_ring=CyclotomicField(8580))
M = H._module
chi = DirichletCharacter(H, M([0,6435,8502,7700]))
pari: [g,chi] = znchar(Mod(61,327184))
Basic properties
| Modulus: | \(327184\) | |
| Conductor: | \(327184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8580\) | 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 327184.yg
\(\chi_{327184}(29,\cdot)\) \(\chi_{327184}(61,\cdot)\) \(\chi_{327184}(237,\cdot)\) \(\chi_{327184}(789,\cdot)\) \(\chi_{327184}(893,\cdot)\) \(\chi_{327184}(997,\cdot)\) \(\chi_{327184}(1069,\cdot)\) \(\chi_{327184}(1173,\cdot)\) \(\chi_{327184}(1381,\cdot)\)
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: | $\Q(\zeta_{8580})$ |
| Fixed field: | Number field defined by a degree 8580 polynomial (not computed) |
Values on generators
\((286287,81797,132497,174241)\) → \((1,-i,e\left(\frac{109}{110}\right),e\left(\frac{35}{39}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 327184 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{571}{780}\right)\) | \(e\left(\frac{441}{2860}\right)\) | \(e\left(\frac{991}{2145}\right)\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{1901}{2145}\right)\) | \(e\left(\frac{2489}{4290}\right)\) | \(e\left(\frac{547}{660}\right)\) | \(e\left(\frac{111}{572}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{441}{1430}\right)\) |
sage: chi.jacobi_sum(n)