![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(661, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([8]))
![Copy content]()
pari:[g,chi] = znchar(Mod(400,661))
| Modulus: | \(661\) | |
| Conductor: | \(661\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(33\) |
![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)
|
\(\chi_{661}(11,\cdot)\)
\(\chi_{661}(20,\cdot)\)
\(\chi_{661}(38,\cdot)\)
\(\chi_{661}(87,\cdot)\)
\(\chi_{661}(99,\cdot)\)
\(\chi_{661}(121,\cdot)\)
\(\chi_{661}(122,\cdot)\)
\(\chi_{661}(180,\cdot)\)
\(\chi_{661}(230,\cdot)\)
\(\chi_{661}(295,\cdot)\)
\(\chi_{661}(298,\cdot)\)
\(\chi_{661}(342,\cdot)\)
\(\chi_{661}(400,\cdot)\)
\(\chi_{661}(428,\cdot)\)
\(\chi_{661}(434,\cdot)\)
\(\chi_{661}(437,\cdot)\)
\(\chi_{661}(547,\cdot)\)
\(\chi_{661}(601,\cdot)\)
\(\chi_{661}(628,\cdot)\)
\(\chi_{661}(632,\cdot)\)
![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 ]
\(2\) → \(e\left(\frac{4}{33}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 661 }(400, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) |
![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)
