![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,4,33]))
![Copy content]()
pari:[g,chi] = znchar(Mod(41,663))
| Modulus: | \(663\) | |
| Conductor: | \(663\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
![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)
|
\(\chi_{663}(20,\cdot)\)
\(\chi_{663}(41,\cdot)\)
\(\chi_{663}(167,\cdot)\)
\(\chi_{663}(197,\cdot)\)
\(\chi_{663}(227,\cdot)\)
\(\chi_{663}(266,\cdot)\)
\(\chi_{663}(284,\cdot)\)
\(\chi_{663}(362,\cdot)\)
\(\chi_{663}(371,\cdot)\)
\(\chi_{663}(431,\cdot)\)
\(\chi_{663}(470,\cdot)\)
\(\chi_{663}(479,\cdot)\)
\(\chi_{663}(488,\cdot)\)
\(\chi_{663}(500,\cdot)\)
\(\chi_{663}(566,\cdot)\)
\(\chi_{663}(656,\cdot)\)
![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 ]
\((443,613,547)\) → \((-1,e\left(\frac{1}{12}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 663 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) |
![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)
