from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,15,12,16]))
pari: [g,chi] = znchar(Mod(53,6048))
Basic properties
| Modulus: | \(6048\) | |
| Conductor: | \(672\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{672}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.hb
\(\chi_{6048}(53,\cdot)\) \(\chi_{6048}(485,\cdot)\) \(\chi_{6048}(1565,\cdot)\) \(\chi_{6048}(1997,\cdot)\) \(\chi_{6048}(3077,\cdot)\) \(\chi_{6048}(3509,\cdot)\) \(\chi_{6048}(4589,\cdot)\) \(\chi_{6048}(5021,\cdot)\)
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
\((4159,3781,3809,2593)\) → \((1,e\left(\frac{5}{8}\right),-1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6048 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)