from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,0,21]))
pari: [g,chi] = znchar(Mod(64,8015))
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 \(\chi_{1145}(64,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8015.dy
\(\chi_{8015}(64,\cdot)\) \(\chi_{8015}(169,\cdot)\) \(\chi_{8015}(414,\cdot)\) \(\chi_{8015}(484,\cdot)\) \(\chi_{8015}(1149,\cdot)\) \(\chi_{8015}(1499,\cdot)\) \(\chi_{8015}(1779,\cdot)\) \(\chi_{8015}(2129,\cdot)\) \(\chi_{8015}(2759,\cdot)\) \(\chi_{8015}(2934,\cdot)\) \(\chi_{8015}(3179,\cdot)\) \(\chi_{8015}(4334,\cdot)\) \(\chi_{8015}(4824,\cdot)\) \(\chi_{8015}(5454,\cdot)\) \(\chi_{8015}(5664,\cdot)\) \(\chi_{8015}(6749,\cdot)\) \(\chi_{8015}(6854,\cdot)\) \(\chi_{8015}(7729,\cdot)\)
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
\((3207,4581,4586)\) → \((-1,1,e\left(\frac{21}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 8015 }(64, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
sage: chi.jacobi_sum(n)