from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8023, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,9]))
pari: [g,chi] = znchar(Mod(638,8023))
Basic properties
| Modulus: | \(8023\) | |
| Conductor: | \(8023\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8023.bt
\(\chi_{8023}(638,\cdot)\) \(\chi_{8023}(2413,\cdot)\) \(\chi_{8023}(4259,\cdot)\) \(\chi_{8023}(4472,\cdot)\) \(\chi_{8023}(5466,\cdot)\) \(\chi_{8023}(5608,\cdot)\) \(\chi_{8023}(6602,\cdot)\) \(\chi_{8023}(6815,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | Number field defined by a degree 16 polynomial |
Values on generators
\((6894,3054)\) → \((-1,e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8023 }(638, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)