![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([136,243]))
![Copy content]()
pari:[g,chi] = znchar(Mod(95,867))
| Modulus: | \(867\) | |
| Conductor: | \(867\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(272\) |
![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)
|
\(\chi_{867}(5,\cdot)\)
\(\chi_{867}(11,\cdot)\)
\(\chi_{867}(14,\cdot)\)
\(\chi_{867}(20,\cdot)\)
\(\chi_{867}(23,\cdot)\)
\(\chi_{867}(29,\cdot)\)
\(\chi_{867}(41,\cdot)\)
\(\chi_{867}(44,\cdot)\)
\(\chi_{867}(56,\cdot)\)
\(\chi_{867}(62,\cdot)\)
\(\chi_{867}(71,\cdot)\)
\(\chi_{867}(74,\cdot)\)
\(\chi_{867}(80,\cdot)\)
\(\chi_{867}(92,\cdot)\)
\(\chi_{867}(95,\cdot)\)
\(\chi_{867}(107,\cdot)\)
\(\chi_{867}(113,\cdot)\)
\(\chi_{867}(116,\cdot)\)
\(\chi_{867}(122,\cdot)\)
\(\chi_{867}(125,\cdot)\)
\(\chi_{867}(143,\cdot)\)
\(\chi_{867}(146,\cdot)\)
\(\chi_{867}(164,\cdot)\)
\(\chi_{867}(167,\cdot)\)
\(\chi_{867}(173,\cdot)\)
\(\chi_{867}(176,\cdot)\)
\(\chi_{867}(182,\cdot)\)
\(\chi_{867}(194,\cdot)\)
\(\chi_{867}(197,\cdot)\)
\(\chi_{867}(209,\cdot)\)
...
![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 ]
\((290,292)\) → \((-1,e\left(\frac{243}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 867 }(95, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{23}{272}\right)\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{99}{136}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{13}{272}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{195}{272}\right)\) | \(e\left(\frac{33}{34}\right)\) |
![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)
