![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(247, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([3,13]))
![Copy content]()
pari:[g,chi] = znchar(Mod(212,247))
| Modulus: | \(247\) | |
| Conductor: | \(247\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(18\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{247}(10,\cdot)\)
\(\chi_{247}(127,\cdot)\)
\(\chi_{247}(147,\cdot)\)
\(\chi_{247}(173,\cdot)\)
\(\chi_{247}(205,\cdot)\)
\(\chi_{247}(212,\cdot)\)
![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 ]
\((210,40)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 247 }(212, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) |
![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)
