from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2432, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,16]))
pari: [g,chi] = znchar(Mod(7,2432))
Basic properties
| Modulus: | \(2432\) | |
| Conductor: | \(1216\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1216}(843,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2432.cd
\(\chi_{2432}(7,\cdot)\) \(\chi_{2432}(87,\cdot)\) \(\chi_{2432}(311,\cdot)\) \(\chi_{2432}(391,\cdot)\) \(\chi_{2432}(615,\cdot)\) \(\chi_{2432}(695,\cdot)\) \(\chi_{2432}(919,\cdot)\) \(\chi_{2432}(999,\cdot)\) \(\chi_{2432}(1223,\cdot)\) \(\chi_{2432}(1303,\cdot)\) \(\chi_{2432}(1527,\cdot)\) \(\chi_{2432}(1607,\cdot)\) \(\chi_{2432}(1831,\cdot)\) \(\chi_{2432}(1911,\cdot)\) \(\chi_{2432}(2135,\cdot)\) \(\chi_{2432}(2215,\cdot)\)
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
\((1407,2053,1921)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 2432 }(7, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) |
sage: chi.jacobi_sum(n)