from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4704, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,1,4,4]))
pari: [g,chi] = znchar(Mod(293,4704))
Basic properties
| Modulus: | \(4704\) | |
| Conductor: | \(672\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{672}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4704.bp
\(\chi_{4704}(293,\cdot)\) \(\chi_{4704}(1469,\cdot)\) \(\chi_{4704}(2645,\cdot)\) \(\chi_{4704}(3821,\cdot)\)
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: | 8.8.417644767346688.4 |
Values on generators
\((1471,1765,3137,4609)\) → \((1,e\left(\frac{1}{8}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4704 }(293, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)