from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360000, base_ring=CyclotomicField(6000))
M = H._module
chi = DirichletCharacter(H, M([3000,2625,1000,2916]))
pari: [g,chi] = znchar(Mod(83,360000))
Basic properties
| Modulus: | \(360000\) | |
| Conductor: | \(360000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6000\) | 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 360000.vy
\(\chi_{360000}(83,\cdot)\) \(\chi_{360000}(347,\cdot)\) \(\chi_{360000}(563,\cdot)\) \(\chi_{360000}(587,\cdot)\) \(\chi_{360000}(803,\cdot)\) \(\chi_{360000}(1067,\cdot)\) \(\chi_{360000}(1283,\cdot)\) \(\chi_{360000}(1523,\cdot)\)
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: | $\Q(\zeta_{6000})$ |
| Fixed field: | Number field defined by a degree 6000 polynomial (not computed) |
Values on generators
\((258751,202501,280001,29377)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{243}{500}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 360000 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{151}{600}\right)\) | \(e\left(\frac{1141}{6000}\right)\) | \(e\left(\frac{2699}{6000}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{1421}{2000}\right)\) | \(e\left(\frac{2773}{3000}\right)\) | \(e\left(\frac{4267}{6000}\right)\) | \(e\left(\frac{248}{375}\right)\) | \(e\left(\frac{1663}{2000}\right)\) | \(e\left(\frac{1027}{3000}\right)\) |
sage: chi.jacobi_sum(n)