from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,5,8]))
pari: [g,chi] = znchar(Mod(207,3328))
Basic properties
| Modulus: | \(3328\) | |
| Conductor: | \(832\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{832}(779,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3328.bz
\(\chi_{3328}(207,\cdot)\) \(\chi_{3328}(623,\cdot)\) \(\chi_{3328}(1039,\cdot)\) \(\chi_{3328}(1455,\cdot)\) \(\chi_{3328}(1871,\cdot)\) \(\chi_{3328}(2287,\cdot)\) \(\chi_{3328}(2703,\cdot)\) \(\chi_{3328}(3119,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.0.493078965234878699415351955816448.1 |
Values on generators
\((1535,261,769)\) → \((-1,e\left(\frac{5}{16}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3328 }(207, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)