from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(580, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,9]))
pari: [g,chi] = znchar(Mod(483,580))
Basic properties
| Modulus: | \(580\) | |
| Conductor: | \(580\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 580.bm
\(\chi_{580}(3,\cdot)\) \(\chi_{580}(27,\cdot)\) \(\chi_{580}(43,\cdot)\) \(\chi_{580}(47,\cdot)\) \(\chi_{580}(243,\cdot)\) \(\chi_{580}(247,\cdot)\) \(\chi_{580}(263,\cdot)\) \(\chi_{580}(287,\cdot)\) \(\chi_{580}(327,\cdot)\) \(\chi_{580}(387,\cdot)\) \(\chi_{580}(483,\cdot)\) \(\chi_{580}(543,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.0.390801221884227140575403105214708756468352000000000000000000000.2 |
Values on generators
\((291,117,321)\) → \((-1,-i,e\left(\frac{9}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 580 }(483, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) |
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)