![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,18,15]))
![Copy content]()
pari:[g,chi] = znchar(Mod(30,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
![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)
|
\(\chi_{1309}(30,\cdot)\)
\(\chi_{1309}(72,\cdot)\)
\(\chi_{1309}(123,\cdot)\)
\(\chi_{1309}(149,\cdot)\)
\(\chi_{1309}(200,\cdot)\)
\(\chi_{1309}(310,\cdot)\)
\(\chi_{1309}(387,\cdot)\)
\(\chi_{1309}(480,\cdot)\)
\(\chi_{1309}(557,\cdot)\)
\(\chi_{1309}(667,\cdot)\)
\(\chi_{1309}(744,\cdot)\)
\(\chi_{1309}(1075,\cdot)\)
\(\chi_{1309}(1152,\cdot)\)
\(\chi_{1309}(1194,\cdot)\)
\(\chi_{1309}(1262,\cdot)\)
\(\chi_{1309}(1271,\cdot)\)
![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 ]
\((1123,596,309)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(30, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{10}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
