from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4160, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,39,24,4]))
pari: [g,chi] = znchar(Mod(3499,4160))
Basic properties
| Modulus: | \(4160\) | |
| Conductor: | \(4160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4160.jz
\(\chi_{4160}(379,\cdot)\) \(\chi_{4160}(539,\cdot)\) \(\chi_{4160}(579,\cdot)\) \(\chi_{4160}(739,\cdot)\) \(\chi_{4160}(1419,\cdot)\) \(\chi_{4160}(1579,\cdot)\) \(\chi_{4160}(1619,\cdot)\) \(\chi_{4160}(1779,\cdot)\) \(\chi_{4160}(2459,\cdot)\) \(\chi_{4160}(2619,\cdot)\) \(\chi_{4160}(2659,\cdot)\) \(\chi_{4160}(2819,\cdot)\) \(\chi_{4160}(3499,\cdot)\) \(\chi_{4160}(3659,\cdot)\) \(\chi_{4160}(3699,\cdot)\) \(\chi_{4160}(3859,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((4031,261,2497,1601)\) → \((-1,e\left(\frac{13}{16}\right),-1,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4160 }(3499, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{48}\right)\) |
sage: chi.jacobi_sum(n)