from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4284, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,8,0,15]))
pari: [g,chi] = znchar(Mod(2983,4284))
Basic properties
| Modulus: | \(4284\) | |
| Conductor: | \(612\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{612}(535,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4284.ga
\(\chi_{4284}(43,\cdot)\) \(\chi_{4284}(967,\cdot)\) \(\chi_{4284}(1471,\cdot)\) \(\chi_{4284}(1555,\cdot)\) \(\chi_{4284}(2059,\cdot)\) \(\chi_{4284}(2983,\cdot)\) \(\chi_{4284}(3487,\cdot)\) \(\chi_{4284}(3823,\cdot)\)
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
\((2143,3809,1837,1261)\) → \((-1,e\left(\frac{1}{3}\right),1,e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4284 }(2983, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) |
sage: chi.jacobi_sum(n)