from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3744, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,0,4]))
pari: [g,chi] = znchar(Mod(667,3744))
Basic properties
| Modulus: | \(3744\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{416}(251,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3744.jz
\(\chi_{3744}(595,\cdot)\) \(\chi_{3744}(667,\cdot)\) \(\chi_{3744}(1531,\cdot)\) \(\chi_{3744}(1603,\cdot)\) \(\chi_{3744}(2467,\cdot)\) \(\chi_{3744}(2539,\cdot)\) \(\chi_{3744}(3403,\cdot)\) \(\chi_{3744}(3475,\cdot)\)
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: | 24.0.188216044816745326913150945765287080795084676923392.1 |
Values on generators
\((703,2341,2081,2017)\) → \((-1,e\left(\frac{1}{8}\right),1,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3744 }(667, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{1}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) |
sage: chi.jacobi_sum(n)