from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(255, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,7]))
pari: [g,chi] = znchar(Mod(11,255))
Basic properties
| Modulus: | \(255\) | |
| Conductor: | \(51\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{51}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 255.bg
\(\chi_{255}(11,\cdot)\) \(\chi_{255}(41,\cdot)\) \(\chi_{255}(56,\cdot)\) \(\chi_{255}(71,\cdot)\) \(\chi_{255}(116,\cdot)\) \(\chi_{255}(131,\cdot)\) \(\chi_{255}(146,\cdot)\) \(\chi_{255}(176,\cdot)\)
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: | \(\Q(\zeta_{51})^+\) |
Values on generators
\((86,52,241)\) → \((-1,1,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 255 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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)