![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(680, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,4,4,5]))
![Copy content]()
pari:[g,chi] = znchar(Mod(229,680))
| Modulus: | \(680\) | |
| Conductor: | \(680\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(8\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{680}(189,\cdot)\)
\(\chi_{680}(229,\cdot)\)
\(\chi_{680}(349,\cdot)\)
\(\chi_{680}(389,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((511,341,137,241)\) → \((1,-1,-1,e\left(\frac{5}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 680 }(229, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(i\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
