from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,7,8,8]))
pari: [g,chi] = znchar(Mod(749,960))
Basic properties
| Modulus: | \(960\) | |
| Conductor: | \(960\) | 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 960.cl
\(\chi_{960}(29,\cdot)\) \(\chi_{960}(149,\cdot)\) \(\chi_{960}(269,\cdot)\) \(\chi_{960}(389,\cdot)\) \(\chi_{960}(509,\cdot)\) \(\chi_{960}(629,\cdot)\) \(\chi_{960}(749,\cdot)\) \(\chi_{960}(869,\cdot)\)
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: | 16.0.1549172324705387112352972800000000.1 |
Values on generators
\((511,901,641,577)\) → \((1,e\left(\frac{7}{16}\right),-1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 960 }(749, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)