from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,16,4]))
pari: [g,chi] = znchar(Mod(4987,6048))
Basic properties
| Modulus: | \(6048\) | |
| Conductor: | \(2016\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2016}(1627,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.hi
\(\chi_{6048}(451,\cdot)\) \(\chi_{6048}(523,\cdot)\) \(\chi_{6048}(1963,\cdot)\) \(\chi_{6048}(2035,\cdot)\) \(\chi_{6048}(3475,\cdot)\) \(\chi_{6048}(3547,\cdot)\) \(\chi_{6048}(4987,\cdot)\) \(\chi_{6048}(5059,\cdot)\)
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: | 24.24.1464301637584529641150268420322112322377688645671104200835072.2 |
Values on generators
\((4159,3781,3809,2593)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6048 }(4987, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(1\) | \(e\left(\frac{11}{24}\right)\) |
sage: chi.jacobi_sum(n)