from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,28,21]))
pari: [g,chi] = znchar(Mod(11,663))
Basic properties
| Modulus: | \(663\) | |
| Conductor: | \(663\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 663.ct
\(\chi_{663}(11,\cdot)\) \(\chi_{663}(71,\cdot)\) \(\chi_{663}(80,\cdot)\) \(\chi_{663}(158,\cdot)\) \(\chi_{663}(176,\cdot)\) \(\chi_{663}(215,\cdot)\) \(\chi_{663}(245,\cdot)\) \(\chi_{663}(275,\cdot)\) \(\chi_{663}(401,\cdot)\) \(\chi_{663}(422,\cdot)\) \(\chi_{663}(449,\cdot)\) \(\chi_{663}(539,\cdot)\) \(\chi_{663}(605,\cdot)\) \(\chi_{663}(617,\cdot)\) \(\chi_{663}(626,\cdot)\) \(\chi_{663}(635,\cdot)\)
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
\((443,613,547)\) → \((-1,e\left(\frac{7}{12}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 663 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) |
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)