from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([6,5]))
pari: [g,chi] = znchar(Mod(1659,6001))
Basic properties
Modulus: | \(6001\) | |
Conductor: | \(6001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 6001.cc
\(\chi_{6001}(346,\cdot)\) \(\chi_{6001}(1049,\cdot)\) \(\chi_{6001}(1659,\cdot)\) \(\chi_{6001}(2538,\cdot)\) \(\chi_{6001}(2577,\cdot)\) \(\chi_{6001}(2608,\cdot)\) \(\chi_{6001}(2723,\cdot)\) \(\chi_{6001}(2765,\cdot)\) \(\chi_{6001}(2834,\cdot)\) \(\chi_{6001}(4230,\cdot)\) \(\chi_{6001}(4295,\cdot)\) \(\chi_{6001}(4596,\cdot)\) \(\chi_{6001}(5158,\cdot)\) \(\chi_{6001}(5301,\cdot)\) \(\chi_{6001}(5581,\cdot)\) \(\chi_{6001}(5749,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((2825,3180)\) → \((e\left(\frac{3}{16}\right),e\left(\frac{5}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6001 }(1659, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{11}{32}\right)\) | \(-1\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)