from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8024, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,5,0]))
pari: [g,chi] = znchar(Mod(1535,8024))
Basic properties
Modulus: | \(8024\) | |
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 \(\chi_{68}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8024.bh
\(\chi_{8024}(1535,\cdot)\) \(\chi_{8024}(2479,\cdot)\) \(\chi_{8024}(2951,\cdot)\) \(\chi_{8024}(3423,\cdot)\) \(\chi_{8024}(4839,\cdot)\) \(\chi_{8024}(5311,\cdot)\) \(\chi_{8024}(5783,\cdot)\) \(\chi_{8024}(6727,\cdot)\)
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
\((2007,4013,3777,3129)\) → \((-1,1,e\left(\frac{5}{16}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 8024 }(1535, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)