from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6664, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,0,7]))
pari: [g,chi] = znchar(Mod(2255,6664))
Basic properties
Modulus: | \(6664\) | |
Conductor: | \(68\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{68}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6664.cx
\(\chi_{6664}(295,\cdot)\) \(\chi_{6664}(687,\cdot)\) \(\chi_{6664}(1863,\cdot)\) \(\chi_{6664}(2255,\cdot)\) \(\chi_{6664}(2647,\cdot)\) \(\chi_{6664}(3431,\cdot)\) \(\chi_{6664}(5783,\cdot)\) \(\chi_{6664}(6567,\cdot)\)
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: | \(\Q(\zeta_{68})^+\) |
Values on generators
\((4999,3333,4217,785)\) → \((-1,1,1,e\left(\frac{7}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 6664 }(2255, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)