from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,0,3]))
pari: [g,chi] = znchar(Mod(176,8015))
Basic properties
| Modulus: | \(8015\) | |
| Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{229}(176,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8015.eb
\(\chi_{8015}(176,\cdot)\) \(\chi_{8015}(526,\cdot)\) \(\chi_{8015}(1156,\cdot)\) \(\chi_{8015}(1331,\cdot)\) \(\chi_{8015}(1576,\cdot)\) \(\chi_{8015}(2731,\cdot)\) \(\chi_{8015}(3221,\cdot)\) \(\chi_{8015}(3851,\cdot)\) \(\chi_{8015}(4061,\cdot)\) \(\chi_{8015}(5146,\cdot)\) \(\chi_{8015}(5251,\cdot)\) \(\chi_{8015}(6126,\cdot)\) \(\chi_{8015}(6476,\cdot)\) \(\chi_{8015}(6581,\cdot)\) \(\chi_{8015}(6826,\cdot)\) \(\chi_{8015}(6896,\cdot)\) \(\chi_{8015}(7561,\cdot)\) \(\chi_{8015}(7911,\cdot)\)
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
\((3207,4581,4586)\) → \((1,1,e\left(\frac{3}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 8015 }(176, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) |
sage: chi.jacobi_sum(n)