from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(687, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,25]))
pari: [g,chi] = znchar(Mod(626,687))
Basic properties
| Modulus: | \(687\) | |
| Conductor: | \(687\) | 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 687.n
\(\chi_{687}(11,\cdot)\) \(\chi_{687}(26,\cdot)\) \(\chi_{687}(68,\cdot)\) \(\chi_{687}(125,\cdot)\) \(\chi_{687}(176,\cdot)\) \(\chi_{687}(185,\cdot)\) \(\chi_{687}(212,\cdot)\) \(\chi_{687}(233,\cdot)\) \(\chi_{687}(293,\cdot)\) \(\chi_{687}(398,\cdot)\) \(\chi_{687}(401,\cdot)\) \(\chi_{687}(416,\cdot)\) \(\chi_{687}(431,\cdot)\) \(\chi_{687}(473,\cdot)\) \(\chi_{687}(566,\cdot)\) \(\chi_{687}(626,\cdot)\) \(\chi_{687}(644,\cdot)\) \(\chi_{687}(671,\cdot)\)
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: | 38.0.2394510171790650820123124474406353844872595054967993341776661644110188322971389918926750746438903.1 |
Values on generators
\((230,235)\) → \((-1,e\left(\frac{25}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 687 }(626, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)