![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([27,100]))
![Copy content]()
pari:[g,chi] = znchar(Mod(283,475))
| Modulus: | \(475\) | |
| Conductor: | \(475\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(180\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{475}(17,\cdot)\)
\(\chi_{475}(23,\cdot)\)
\(\chi_{475}(28,\cdot)\)
\(\chi_{475}(42,\cdot)\)
\(\chi_{475}(47,\cdot)\)
\(\chi_{475}(62,\cdot)\)
\(\chi_{475}(63,\cdot)\)
\(\chi_{475}(73,\cdot)\)
\(\chi_{475}(92,\cdot)\)
\(\chi_{475}(112,\cdot)\)
\(\chi_{475}(123,\cdot)\)
\(\chi_{475}(137,\cdot)\)
\(\chi_{475}(138,\cdot)\)
\(\chi_{475}(142,\cdot)\)
\(\chi_{475}(158,\cdot)\)
\(\chi_{475}(177,\cdot)\)
\(\chi_{475}(187,\cdot)\)
\(\chi_{475}(188,\cdot)\)
\(\chi_{475}(213,\cdot)\)
\(\chi_{475}(233,\cdot)\)
\(\chi_{475}(237,\cdot)\)
\(\chi_{475}(252,\cdot)\)
\(\chi_{475}(253,\cdot)\)
\(\chi_{475}(263,\cdot)\)
\(\chi_{475}(272,\cdot)\)
\(\chi_{475}(283,\cdot)\)
\(\chi_{475}(302,\cdot)\)
\(\chi_{475}(308,\cdot)\)
\(\chi_{475}(313,\cdot)\)
\(\chi_{475}(327,\cdot)\)
...
![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 ]
\((77,401)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(283, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{49}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{113}{180}\right)\) |
![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)
