![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,20]))
![Copy content]()
pari:[g,chi] = znchar(Mod(19,224))
| Modulus: | \(224\) | |
| Conductor: | \(224\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(24\) |
![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)
|
\(\chi_{224}(3,\cdot)\)
\(\chi_{224}(19,\cdot)\)
\(\chi_{224}(59,\cdot)\)
\(\chi_{224}(75,\cdot)\)
\(\chi_{224}(115,\cdot)\)
\(\chi_{224}(131,\cdot)\)
\(\chi_{224}(171,\cdot)\)
\(\chi_{224}(187,\cdot)\)
![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 ]
\((127,197,129)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 224 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) |
![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)
