![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(511, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([60,43]))
![Copy content]()
pari:[g,chi] = znchar(Mod(131,511))
| Modulus: | \(511\) | |
| Conductor: | \(511\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(72\) |
![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)
|
\(\chi_{511}(5,\cdot)\)
\(\chi_{511}(26,\cdot)\)
\(\chi_{511}(31,\cdot)\)
\(\chi_{511}(33,\cdot)\)
\(\chi_{511}(40,\cdot)\)
\(\chi_{511}(47,\cdot)\)
\(\chi_{511}(59,\cdot)\)
\(\chi_{511}(68,\cdot)\)
\(\chi_{511}(87,\cdot)\)
\(\chi_{511}(115,\cdot)\)
\(\chi_{511}(131,\cdot)\)
\(\chi_{511}(185,\cdot)\)
\(\chi_{511}(199,\cdot)\)
\(\chi_{511}(206,\cdot)\)
\(\chi_{511}(208,\cdot)\)
\(\chi_{511}(248,\cdot)\)
\(\chi_{511}(264,\cdot)\)
\(\chi_{511}(320,\cdot)\)
\(\chi_{511}(376,\cdot)\)
\(\chi_{511}(409,\cdot)\)
\(\chi_{511}(451,\cdot)\)
\(\chi_{511}(453,\cdot)\)
\(\chi_{511}(458,\cdot)\)
\(\chi_{511}(472,\cdot)\)
![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 ]
\((220,78)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{43}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 511 }(131, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) |
![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)
