from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6031, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([36,52]))
pari: [g,chi] = znchar(Mod(787,6031))
Basic properties
| Modulus: | \(6031\) | |
| Conductor: | \(6031\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | 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.de
\(\chi_{6031}(787,\cdot)\) \(\chi_{6031}(951,\cdot)\) \(\chi_{6031}(1136,\cdot)\) \(\chi_{6031}(1617,\cdot)\) \(\chi_{6031}(1691,\cdot)\) \(\chi_{6031}(2304,\cdot)\) \(\chi_{6031}(2600,\cdot)\) \(\chi_{6031}(3060,\cdot)\) \(\chi_{6031}(3118,\cdot)\) \(\chi_{6031}(3266,\cdot)\) \(\chi_{6031}(3488,\cdot)\) \(\chi_{6031}(3895,\cdot)\) \(\chi_{6031}(3948,\cdot)\) \(\chi_{6031}(4207,\cdot)\) \(\chi_{6031}(5280,\cdot)\) \(\chi_{6031}(5782,\cdot)\) \(\chi_{6031}(5893,\cdot)\) \(\chi_{6031}(5983,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 27 polynomial |
Values on generators
\((816,5218)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{26}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6031 }(787, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) |
sage: chi.jacobi_sum(n)