from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,24,24,32,21]))
pari: [g,chi] = znchar(Mod(11,2856))
Basic properties
| Modulus: | \(2856\) | |
| Conductor: | \(2856\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2856.gd
\(\chi_{2856}(11,\cdot)\) \(\chi_{2856}(107,\cdot)\) \(\chi_{2856}(275,\cdot)\) \(\chi_{2856}(347,\cdot)\) \(\chi_{2856}(515,\cdot)\) \(\chi_{2856}(683,\cdot)\) \(\chi_{2856}(779,\cdot)\) \(\chi_{2856}(947,\cdot)\) \(\chi_{2856}(1115,\cdot)\) \(\chi_{2856}(1187,\cdot)\) \(\chi_{2856}(1355,\cdot)\) \(\chi_{2856}(1451,\cdot)\) \(\chi_{2856}(1523,\cdot)\) \(\chi_{2856}(1859,\cdot)\) \(\chi_{2856}(2459,\cdot)\) \(\chi_{2856}(2795,\cdot)\)
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
\((2143,1429,953,409,2689)\) → \((-1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)