![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,24,9]))
![Copy content]()
pari:[g,chi] = znchar(Mod(10,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
![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)
|
\(\chi_{1309}(10,\cdot)\)
\(\chi_{1309}(54,\cdot)\)
\(\chi_{1309}(131,\cdot)\)
\(\chi_{1309}(164,\cdot)\)
\(\chi_{1309}(241,\cdot)\)
\(\chi_{1309}(318,\cdot)\)
\(\chi_{1309}(362,\cdot)\)
\(\chi_{1309}(439,\cdot)\)
\(\chi_{1309}(516,\cdot)\)
\(\chi_{1309}(549,\cdot)\)
\(\chi_{1309}(626,\cdot)\)
\(\chi_{1309}(670,\cdot)\)
\(\chi_{1309}(703,\cdot)\)
\(\chi_{1309}(857,\cdot)\)
\(\chi_{1309}(1132,\cdot)\)
\(\chi_{1309}(1286,\cdot)\)
![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 ]
\((1123,596,309)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(10, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-i\) |
![Copy content]()
sage:chi.jacobi_sum(n)
