from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(896, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,8]))
pari: [g,chi] = znchar(Mod(73,896))
Basic properties
Modulus: | \(896\) | |
Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{448}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 896.bo
\(\chi_{896}(73,\cdot)\) \(\chi_{896}(89,\cdot)\) \(\chi_{896}(185,\cdot)\) \(\chi_{896}(201,\cdot)\) \(\chi_{896}(297,\cdot)\) \(\chi_{896}(313,\cdot)\) \(\chi_{896}(409,\cdot)\) \(\chi_{896}(425,\cdot)\) \(\chi_{896}(521,\cdot)\) \(\chi_{896}(537,\cdot)\) \(\chi_{896}(633,\cdot)\) \(\chi_{896}(649,\cdot)\) \(\chi_{896}(745,\cdot)\) \(\chi_{896}(761,\cdot)\) \(\chi_{896}(857,\cdot)\) \(\chi_{896}(873,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,645,129)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 896 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) |
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)