from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1472, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,11,8]))
pari: [g,chi] = znchar(Mod(413,1472))
Basic properties
| Modulus: | \(1472\) | |
| Conductor: | \(1472\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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 1472.s
\(\chi_{1472}(45,\cdot)\) \(\chi_{1472}(229,\cdot)\) \(\chi_{1472}(413,\cdot)\) \(\chi_{1472}(597,\cdot)\) \(\chi_{1472}(781,\cdot)\) \(\chi_{1472}(965,\cdot)\) \(\chi_{1472}(1149,\cdot)\) \(\chi_{1472}(1333,\cdot)\)
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: | 16.0.47336086032831043196344263117897728.1 |
Values on generators
\((1151,645,833)\) → \((1,e\left(\frac{11}{16}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1472 }(413, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)