from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3400, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,4,4,7]))
pari: [g,chi] = znchar(Mod(699,3400))
Basic properties
| Modulus: | \(3400\) | |
| Conductor: | \(680\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{680}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3400.br
\(\chi_{3400}(699,\cdot)\) \(\chi_{3400}(899,\cdot)\) \(\chi_{3400}(2099,\cdot)\) \(\chi_{3400}(2899,\cdot)\)
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: | 8.0.1050467002880000.6 |
Values on generators
\((2551,1701,2177,1601)\) → \((-1,-1,-1,e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3400 }(699, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(i\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)