![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(715, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,18,20]))
![Copy content]()
pari:[g,chi] = znchar(Mod(9,715))
| Modulus: | \(715\) | |
| Conductor: | \(715\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(30\) |
![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)
|
\(\chi_{715}(9,\cdot)\)
\(\chi_{715}(159,\cdot)\)
\(\chi_{715}(224,\cdot)\)
\(\chi_{715}(269,\cdot)\)
\(\chi_{715}(289,\cdot)\)
\(\chi_{715}(334,\cdot)\)
\(\chi_{715}(399,\cdot)\)
\(\chi_{715}(614,\cdot)\)
![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 ]
\((287,651,496)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 715 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
![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)
