![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,24,15]))
![Copy content]()
pari:[g,chi] = znchar(Mod(1257,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
![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)
|
\(\chi_{1309}(16,\cdot)\)
\(\chi_{1309}(135,\cdot)\)
\(\chi_{1309}(543,\cdot)\)
\(\chi_{1309}(730,\cdot)\)
\(\chi_{1309}(900,\cdot)\)
\(\chi_{1309}(1087,\cdot)\)
\(\chi_{1309}(1138,\cdot)\)
\(\chi_{1309}(1257,\cdot)\)
![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 ]
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1257, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
