![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(433, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([4]))
![Copy content]()
pari:[g,chi] = znchar(Mod(289,433))
| Modulus: | \(433\) | |
| Conductor: | \(433\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(27\) |
![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)
|
\(\chi_{433}(3,\cdot)\)
\(\chi_{433}(9,\cdot)\)
\(\chi_{433}(17,\cdot)\)
\(\chi_{433}(22,\cdot)\)
\(\chi_{433}(26,\cdot)\)
\(\chi_{433}(50,\cdot)\)
\(\chi_{433}(51,\cdot)\)
\(\chi_{433}(66,\cdot)\)
\(\chi_{433}(78,\cdot)\)
\(\chi_{433}(81,\cdot)\)
\(\chi_{433}(139,\cdot)\)
\(\chi_{433}(161,\cdot)\)
\(\chi_{433}(243,\cdot)\)
\(\chi_{433}(269,\cdot)\)
\(\chi_{433}(289,\cdot)\)
\(\chi_{433}(335,\cdot)\)
\(\chi_{433}(374,\cdot)\)
\(\chi_{433}(385,\cdot)\)
![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 ]
\(5\) → \(e\left(\frac{2}{27}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 433 }(289, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) |
![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)
