from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(483, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,30]))
pari: [g,chi] = znchar(Mod(32,483))
Basic properties
Modulus: | \(483\) | |
Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 483.z
\(\chi_{483}(2,\cdot)\) \(\chi_{483}(32,\cdot)\) \(\chi_{483}(95,\cdot)\) \(\chi_{483}(128,\cdot)\) \(\chi_{483}(170,\cdot)\) \(\chi_{483}(179,\cdot)\) \(\chi_{483}(200,\cdot)\) \(\chi_{483}(233,\cdot)\) \(\chi_{483}(242,\cdot)\) \(\chi_{483}(284,\cdot)\) \(\chi_{483}(305,\cdot)\) \(\chi_{483}(317,\cdot)\) \(\chi_{483}(326,\cdot)\) \(\chi_{483}(338,\cdot)\) \(\chi_{483}(347,\cdot)\) \(\chi_{483}(380,\cdot)\) \(\chi_{483}(422,\cdot)\) \(\chi_{483}(443,\cdot)\) \(\chi_{483}(464,\cdot)\) \(\chi_{483}(473,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((323,346,442)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 483 }(32, a) \) | \(-1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) |
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)