from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,0,6]))
pari: [g,chi] = znchar(Mod(519,8015))
Basic properties
| Modulus: | \(8015\) | |
| Conductor: | \(1145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1145}(519,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8015.ed
\(\chi_{8015}(519,\cdot)\) \(\chi_{8015}(729,\cdot)\) \(\chi_{8015}(1359,\cdot)\) \(\chi_{8015}(1849,\cdot)\) \(\chi_{8015}(3004,\cdot)\) \(\chi_{8015}(3249,\cdot)\) \(\chi_{8015}(3424,\cdot)\) \(\chi_{8015}(4054,\cdot)\) \(\chi_{8015}(4404,\cdot)\) \(\chi_{8015}(4684,\cdot)\) \(\chi_{8015}(5034,\cdot)\) \(\chi_{8015}(5699,\cdot)\) \(\chi_{8015}(5769,\cdot)\) \(\chi_{8015}(6014,\cdot)\) \(\chi_{8015}(6119,\cdot)\) \(\chi_{8015}(6469,\cdot)\) \(\chi_{8015}(7344,\cdot)\) \(\chi_{8015}(7449,\cdot)\)
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: | Number field defined by a degree 38 polynomial |
Values on generators
\((3207,4581,4586)\) → \((-1,1,e\left(\frac{3}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 8015 }(519, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) |
sage: chi.jacobi_sum(n)