from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,8]))
pari: [g,chi] = znchar(Mod(13,1344))
Basic properties
| Modulus: | \(1344\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1344.ck
\(\chi_{1344}(13,\cdot)\) \(\chi_{1344}(181,\cdot)\) \(\chi_{1344}(349,\cdot)\) \(\chi_{1344}(517,\cdot)\) \(\chi_{1344}(685,\cdot)\) \(\chi_{1344}(853,\cdot)\) \(\chi_{1344}(1021,\cdot)\) \(\chi_{1344}(1189,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.0.3484608386920116940487669055488.4 |
Values on generators
\((127,1093,449,577)\) → \((1,e\left(\frac{15}{16}\right),1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1344 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)