from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4928, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,1,4,4]))
pari: [g,chi] = znchar(Mod(153,4928))
Basic properties
Modulus: | \(4928\) | |
Conductor: | \(2464\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2464}(2309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4928.bp
\(\chi_{4928}(153,\cdot)\) \(\chi_{4928}(1385,\cdot)\) \(\chi_{4928}(2617,\cdot)\) \(\chi_{4928}(3849,\cdot)\)
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: | 8.8.75490580724973568.1 |
Values on generators
\((4159,1541,2817,3137)\) → \((1,e\left(\frac{1}{8}\right),-1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 4928 }(153, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)