from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,14,4]))
pari: [g,chi] = znchar(Mod(428,2365))
Basic properties
Modulus: | \(2365\) | |
Conductor: | \(2365\) | 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 2365.bu
\(\chi_{2365}(428,\cdot)\) \(\chi_{2365}(527,\cdot)\) \(\chi_{2365}(637,\cdot)\) \(\chi_{2365}(692,\cdot)\) \(\chi_{2365}(747,\cdot)\) \(\chi_{2365}(967,\cdot)\) \(\chi_{2365}(1473,\cdot)\) \(\chi_{2365}(1583,\cdot)\) \(\chi_{2365}(1638,\cdot)\) \(\chi_{2365}(1693,\cdot)\) \(\chi_{2365}(1847,\cdot)\) \(\chi_{2365}(1913,\cdot)\)
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
\((947,431,1981)\) → \((-i,-1,e\left(\frac{1}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 2365 }(428, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)