from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(764, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,34]))
pari: [g,chi] = znchar(Mod(503,764))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 764.k
\(\chi_{764}(107,\cdot)\) \(\chi_{764}(223,\cdot)\) \(\chi_{764}(227,\cdot)\) \(\chi_{764}(243,\cdot)\) \(\chi_{764}(327,\cdot)\) \(\chi_{764}(351,\cdot)\) \(\chi_{764}(371,\cdot)\) \(\chi_{764}(387,\cdot)\) \(\chi_{764}(407,\cdot)\) \(\chi_{764}(451,\cdot)\) \(\chi_{764}(503,\cdot)\) \(\chi_{764}(507,\cdot)\) \(\chi_{764}(535,\cdot)\) \(\chi_{764}(559,\cdot)\) \(\chi_{764}(579,\cdot)\) \(\chi_{764}(603,\cdot)\) \(\chi_{764}(723,\cdot)\) \(\chi_{764}(727,\cdot)\)
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: | 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1 |
Values on generators
\((383,401)\) → \((-1,e\left(\frac{17}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 764 }(503, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(-1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) |
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)