![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(816, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,4,8,7]))
![Copy content]()
pari:[g,chi] = znchar(Mod(725,816))
| Modulus: | \(816\) | |
| Conductor: | \(816\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(16\) |
![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)
|
\(\chi_{816}(29,\cdot)\)
\(\chi_{816}(173,\cdot)\)
\(\chi_{816}(197,\cdot)\)
\(\chi_{816}(245,\cdot)\)
\(\chi_{816}(269,\cdot)\)
\(\chi_{816}(413,\cdot)\)
\(\chi_{816}(533,\cdot)\)
\(\chi_{816}(725,\cdot)\)
![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 ]
\((511,613,545,241)\) → \((1,i,-1,e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 816 }(725, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) |
![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)
