![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,24]))
![Copy content]()
pari:[g,chi] = znchar(Mod(131,507))
| Modulus: | \(507\) | |
| Conductor: | \(507\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(26\) |
![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)
|
\(\chi_{507}(14,\cdot)\)
\(\chi_{507}(53,\cdot)\)
\(\chi_{507}(92,\cdot)\)
\(\chi_{507}(131,\cdot)\)
\(\chi_{507}(209,\cdot)\)
\(\chi_{507}(248,\cdot)\)
\(\chi_{507}(287,\cdot)\)
\(\chi_{507}(326,\cdot)\)
\(\chi_{507}(365,\cdot)\)
\(\chi_{507}(404,\cdot)\)
\(\chi_{507}(443,\cdot)\)
\(\chi_{507}(482,\cdot)\)
![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 ]
\((170,340)\) → \((-1,e\left(\frac{12}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(131, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
![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)
