from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3040, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,7,4,4]))
pari: [g,chi] = znchar(Mod(1709,3040))
Basic properties
Modulus: | \(3040\) | |
Conductor: | \(3040\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | 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 3040.cn
\(\chi_{3040}(189,\cdot)\) \(\chi_{3040}(949,\cdot)\) \(\chi_{3040}(1709,\cdot)\) \(\chi_{3040}(2469,\cdot)\)
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: | 8.0.174913885306880000.1 |
Values on generators
\((191,2661,1217,1921)\) → \((1,e\left(\frac{7}{8}\right),-1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3040 }(1709, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)