from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3920, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,21,6]))
pari: [g,chi] = znchar(Mod(83,3920))
Basic properties
Modulus: | \(3920\) | |
Conductor: | \(3920\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3920.eo
\(\chi_{3920}(27,\cdot)\) \(\chi_{3920}(83,\cdot)\) \(\chi_{3920}(643,\cdot)\) \(\chi_{3920}(1147,\cdot)\) \(\chi_{3920}(1203,\cdot)\) \(\chi_{3920}(1707,\cdot)\) \(\chi_{3920}(2267,\cdot)\) \(\chi_{3920}(2323,\cdot)\) \(\chi_{3920}(2827,\cdot)\) \(\chi_{3920}(2883,\cdot)\) \(\chi_{3920}(3387,\cdot)\) \(\chi_{3920}(3443,\cdot)\)
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: | 28.0.129593063934491976647600951409586130589894568495749019992064000000000000000000000.1 |
Values on generators
\((1471,981,3137,3041)\) → \((-1,-i,-i,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 3920 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)