from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8381, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,24]))
pari: [g,chi] = znchar(Mod(5240,8381))
Basic properties
| Modulus: | \(8381\) | |
| Conductor: | \(493\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{493}(310,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8381.bc
\(\chi_{8381}(616,\cdot)\) \(\chi_{8381}(1118,\cdot)\) \(\chi_{8381}(3719,\cdot)\) \(\chi_{8381}(3795,\cdot)\) \(\chi_{8381}(4084,\cdot)\) \(\chi_{8381}(4373,\cdot)\) \(\chi_{8381}(5240,\cdot)\) \(\chi_{8381}(6396,\cdot)\) \(\chi_{8381}(6898,\cdot)\) \(\chi_{8381}(7187,\cdot)\) \(\chi_{8381}(7476,\cdot)\) \(\chi_{8381}(8343,\cdot)\)
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
\((581,8093)\) → \((-i,e\left(\frac{6}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8381 }(5240, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
sage: chi.jacobi_sum(n)