from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,0,3]))
pari: [g,chi] = znchar(Mod(1520,6027))
Basic properties
Modulus: | \(6027\) | |
Conductor: | \(123\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{123}(44,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.z
\(\chi_{6027}(1520,\cdot)\) \(\chi_{6027}(1667,\cdot)\) \(\chi_{6027}(2843,\cdot)\) \(\chi_{6027}(2990,\cdot)\)
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: | 8.8.15775096184361.1 |
Values on generators
\((4019,493,2794)\) → \((-1,1,e\left(\frac{3}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 6027 }(1520, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)