from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1009, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(139,1009))
Basic properties
| Modulus: | \(1009\) | |
| Conductor: | \(1009\) | 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 1009.o
\(\chi_{1009}(139,\cdot)\) \(\chi_{1009}(196,\cdot)\) \(\chi_{1009}(280,\cdot)\) \(\chi_{1009}(339,\cdot)\) \(\chi_{1009}(378,\cdot)\) \(\chi_{1009}(400,\cdot)\) \(\chi_{1009}(609,\cdot)\) \(\chi_{1009}(631,\cdot)\) \(\chi_{1009}(670,\cdot)\) \(\chi_{1009}(729,\cdot)\) \(\chi_{1009}(813,\cdot)\) \(\chi_{1009}(870,\cdot)\)
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
\(11\) → \(e\left(\frac{19}{28}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1009 }(139, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) |
sage: chi.jacobi_sum(n)