from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,7,0,2,0]))
pari: [g,chi] = znchar(Mod(2647,6720))
Basic properties
Modulus: | \(6720\) | |
Conductor: | \(160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{160}(147,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6720.es
\(\chi_{6720}(2647,\cdot)\) \(\chi_{6720}(2983,\cdot)\) \(\chi_{6720}(6007,\cdot)\) \(\chi_{6720}(6343,\cdot)\)
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: | 8.8.33554432000000.2 |
Values on generators
\((1471,3781,4481,5377,1921)\) → \((-1,e\left(\frac{7}{8}\right),1,i,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 6720 }(2647, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)