![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(451, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([4,19]))
![Copy content]()
pari:[g,chi] = znchar(Mod(321,451))
| Modulus: | \(451\) | |
| Conductor: | \(451\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
![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)
|
\(\chi_{451}(6,\cdot)\)
\(\chi_{451}(28,\cdot)\)
\(\chi_{451}(101,\cdot)\)
\(\chi_{451}(106,\cdot)\)
\(\chi_{451}(145,\cdot)\)
\(\chi_{451}(194,\cdot)\)
\(\chi_{451}(216,\cdot)\)
\(\chi_{451}(217,\cdot)\)
\(\chi_{451}(259,\cdot)\)
\(\chi_{451}(261,\cdot)\)
\(\chi_{451}(272,\cdot)\)
\(\chi_{451}(281,\cdot)\)
\(\chi_{451}(304,\cdot)\)
\(\chi_{451}(316,\cdot)\)
\(\chi_{451}(321,\cdot)\)
\(\chi_{451}(376,\cdot)\)
![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 ]
\((288,375)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{19}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 451 }(321, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) |
![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)
