from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,16,32]))
pari: [g,chi] = znchar(Mod(67,4032))
Basic properties
Modulus: | \(4032\) | |
Conductor: | \(4032\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.gs
\(\chi_{4032}(67,\cdot)\) \(\chi_{4032}(331,\cdot)\) \(\chi_{4032}(571,\cdot)\) \(\chi_{4032}(835,\cdot)\) \(\chi_{4032}(1075,\cdot)\) \(\chi_{4032}(1339,\cdot)\) \(\chi_{4032}(1579,\cdot)\) \(\chi_{4032}(1843,\cdot)\) \(\chi_{4032}(2083,\cdot)\) \(\chi_{4032}(2347,\cdot)\) \(\chi_{4032}(2587,\cdot)\) \(\chi_{4032}(2851,\cdot)\) \(\chi_{4032}(3091,\cdot)\) \(\chi_{4032}(3355,\cdot)\) \(\chi_{4032}(3595,\cdot)\) \(\chi_{4032}(3859,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,3781,1793,577)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{48}\right)\) |
sage: chi.jacobi_sum(n)