from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,2,21]))
pari: [g,chi] = znchar(Mod(419,8036))
Basic properties
| Modulus: | \(8036\) | |
| Conductor: | \(8036\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 8036.cd
\(\chi_{8036}(419,\cdot)\) \(\chi_{8036}(811,\cdot)\) \(\chi_{8036}(2715,\cdot)\) \(\chi_{8036}(3107,\cdot)\) \(\chi_{8036}(3863,\cdot)\) \(\chi_{8036}(4255,\cdot)\) \(\chi_{8036}(5011,\cdot)\) \(\chi_{8036}(5403,\cdot)\) \(\chi_{8036}(6159,\cdot)\) \(\chi_{8036}(6551,\cdot)\) \(\chi_{8036}(7307,\cdot)\) \(\chi_{8036}(7699,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((4019,493,785)\) → \((-1,e\left(\frac{1}{14}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 8036 }(419, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)