from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6031, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([1,6]))
pari: [g,chi] = znchar(Mod(594,6031))
Basic properties
| Modulus: | \(6031\) | |
| Conductor: | \(6031\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 6031.dq
\(\chi_{6031}(594,\cdot)\) \(\chi_{6031}(757,\cdot)\) \(\chi_{6031}(1689,\cdot)\) \(\chi_{6031}(2178,\cdot)\) \(\chi_{6031}(2830,\cdot)\) \(\chi_{6031}(3202,\cdot)\) \(\chi_{6031}(3365,\cdot)\) \(\chi_{6031}(3645,\cdot)\) \(\chi_{6031}(4297,\cdot)\) \(\chi_{6031}(4786,\cdot)\) \(\chi_{6031}(4832,\cdot)\) \(\chi_{6031}(5158,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((816,5218)\) → \((e\left(\frac{1}{36}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6031 }(594, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(-i\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)