from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,18,15]))
pari: [g,chi] = znchar(Mod(1007,1309))
Basic properties
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1309.bp
\(\chi_{1309}(13,\cdot)\) \(\chi_{1309}(293,\cdot)\) \(\chi_{1309}(370,\cdot)\) \(\chi_{1309}(888,\cdot)\) \(\chi_{1309}(965,\cdot)\) \(\chi_{1309}(1007,\cdot)\) \(\chi_{1309}(1084,\cdot)\) \(\chi_{1309}(1245,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((1123,596,309)\) → \((-1,e\left(\frac{9}{10}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1007, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)