from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,0,6,8]))
pari: [g,chi] = znchar(Mod(247,6720))
Basic properties
| Modulus: | \(6720\) | |
| Conductor: | \(1120\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1120}(387,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6720.jl
\(\chi_{6720}(247,\cdot)\) \(\chi_{6720}(583,\cdot)\) \(\chi_{6720}(2167,\cdot)\) \(\chi_{6720}(2503,\cdot)\) \(\chi_{6720}(3607,\cdot)\) \(\chi_{6720}(3943,\cdot)\) \(\chi_{6720}(5527,\cdot)\) \(\chi_{6720}(5863,\cdot)\)
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
\((1471,3781,4481,5377,1921)\) → \((-1,e\left(\frac{3}{8}\right),1,i,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6720 }(247, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)