from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,3,2]))
pari: [g,chi] = znchar(Mod(157,160))
Basic properties
Modulus: | \(160\) | |
Conductor: | \(160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | 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 160.bb
\(\chi_{160}(53,\cdot)\) \(\chi_{160}(77,\cdot)\) \(\chi_{160}(133,\cdot)\) \(\chi_{160}(157,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
Fixed field: | 8.0.33554432000000.2 |
Values on generators
\((31,101,97)\) → \((1,e\left(\frac{3}{8}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 160 }(157, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) |
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)