from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,33,0,8]))
pari: [g,chi] = znchar(Mod(1123,1344))
Basic properties
| Modulus: | \(1344\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1344.cy
\(\chi_{1344}(19,\cdot)\) \(\chi_{1344}(115,\cdot)\) \(\chi_{1344}(187,\cdot)\) \(\chi_{1344}(283,\cdot)\) \(\chi_{1344}(355,\cdot)\) \(\chi_{1344}(451,\cdot)\) \(\chi_{1344}(523,\cdot)\) \(\chi_{1344}(619,\cdot)\) \(\chi_{1344}(691,\cdot)\) \(\chi_{1344}(787,\cdot)\) \(\chi_{1344}(859,\cdot)\) \(\chi_{1344}(955,\cdot)\) \(\chi_{1344}(1027,\cdot)\) \(\chi_{1344}(1123,\cdot)\) \(\chi_{1344}(1195,\cdot)\) \(\chi_{1344}(1291,\cdot)\)
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
\((127,1093,449,577)\) → \((-1,e\left(\frac{11}{16}\right),1,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1344 }(1123, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{48}\right)\) |
sage: chi.jacobi_sum(n)