from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3072, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,3,4]))
pari: [g,chi] = znchar(Mod(641,3072))
Basic properties
| Modulus: | \(3072\) | |
| Conductor: | \(96\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{96}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3072.p
\(\chi_{3072}(641,\cdot)\) \(\chi_{3072}(1409,\cdot)\) \(\chi_{3072}(2177,\cdot)\) \(\chi_{3072}(2945,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.0.173946175488.1 |
Values on generators
\((2047,2053,1025)\) → \((1,e\left(\frac{3}{8}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3072 }(641, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)