from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,3,20,4]))
pari: [g,chi] = znchar(Mod(1433,8512))
Basic properties
Modulus: | \(8512\) | |
Conductor: | \(4256\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4256}(901,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.hq
\(\chi_{8512}(297,\cdot)\) \(\chi_{8512}(1433,\cdot)\) \(\chi_{8512}(2425,\cdot)\) \(\chi_{8512}(3561,\cdot)\) \(\chi_{8512}(4553,\cdot)\) \(\chi_{8512}(5689,\cdot)\) \(\chi_{8512}(6681,\cdot)\) \(\chi_{8512}(7817,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((5055,6917,7297,3137)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 8512 }(1433, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-i\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)