![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(763, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([36,20]))
![Copy content]()
pari:[g,chi] = znchar(Mod(116,763))
| Modulus: | \(763\) | |
| Conductor: | \(763\) |
![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)
|
\(\chi_{763}(9,\cdot)\)
\(\chi_{763}(81,\cdot)\)
\(\chi_{763}(116,\cdot)\)
\(\chi_{763}(221,\cdot)\)
\(\chi_{763}(233,\cdot)\)
\(\chi_{763}(240,\cdot)\)
\(\chi_{763}(296,\cdot)\)
\(\chi_{763}(298,\cdot)\)
\(\chi_{763}(352,\cdot)\)
\(\chi_{763}(375,\cdot)\)
\(\chi_{763}(424,\cdot)\)
\(\chi_{763}(457,\cdot)\)
\(\chi_{763}(471,\cdot)\)
\(\chi_{763}(485,\cdot)\)
\(\chi_{763}(550,\cdot)\)
\(\chi_{763}(571,\cdot)\)
\(\chi_{763}(618,\cdot)\)
\(\chi_{763}(634,\cdot)\)
![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 ]
\((437,442)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{10}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 763 }(116, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) |
![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)
