from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1083, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,24]))
pari: [g,chi] = znchar(Mod(1046,1083))
Basic properties
| Modulus: | \(1083\) | |
| Conductor: | \(1083\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | 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 1083.p
\(\chi_{1083}(20,\cdot)\) \(\chi_{1083}(77,\cdot)\) \(\chi_{1083}(134,\cdot)\) \(\chi_{1083}(191,\cdot)\) \(\chi_{1083}(248,\cdot)\) \(\chi_{1083}(305,\cdot)\) \(\chi_{1083}(419,\cdot)\) \(\chi_{1083}(476,\cdot)\) \(\chi_{1083}(533,\cdot)\) \(\chi_{1083}(590,\cdot)\) \(\chi_{1083}(647,\cdot)\) \(\chi_{1083}(704,\cdot)\) \(\chi_{1083}(761,\cdot)\) \(\chi_{1083}(818,\cdot)\) \(\chi_{1083}(875,\cdot)\) \(\chi_{1083}(932,\cdot)\) \(\chi_{1083}(989,\cdot)\) \(\chi_{1083}(1046,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.136635360908492439649635218195335734184282612187547808802428015665989693822130747061222659537193769787.1 |
Values on generators
\((362,724)\) → \((-1,e\left(\frac{12}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1083 }(1046, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) |
sage: chi.jacobi_sum(n)