from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(764, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,33]))
pari: [g,chi] = znchar(Mod(423,764))
Basic properties
| Modulus: | \(764\) | |
| Conductor: | \(764\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 764.l
\(\chi_{764}(11,\cdot)\) \(\chi_{764}(31,\cdot)\) \(\chi_{764}(55,\cdot)\) \(\chi_{764}(139,\cdot)\) \(\chi_{764}(155,\cdot)\) \(\chi_{764}(159,\cdot)\) \(\chi_{764}(275,\cdot)\) \(\chi_{764}(419,\cdot)\) \(\chi_{764}(423,\cdot)\) \(\chi_{764}(543,\cdot)\) \(\chi_{764}(567,\cdot)\) \(\chi_{764}(587,\cdot)\) \(\chi_{764}(611,\cdot)\) \(\chi_{764}(639,\cdot)\) \(\chi_{764}(643,\cdot)\) \(\chi_{764}(695,\cdot)\) \(\chi_{764}(739,\cdot)\) \(\chi_{764}(759,\cdot)\)
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: | 38.38.687661045808093482376579225097085116287670624782479569073265850359469722789446885178751177457664.1 |
Values on generators
\((383,401)\) → \((-1,e\left(\frac{33}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 764 }(423, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) |
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)