from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(483, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,22,63]))
pari: [g,chi] = znchar(Mod(37,483))
Basic properties
| Modulus: | \(483\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 483.bd
\(\chi_{483}(37,\cdot)\) \(\chi_{483}(67,\cdot)\) \(\chi_{483}(79,\cdot)\) \(\chi_{483}(88,\cdot)\) \(\chi_{483}(109,\cdot)\) \(\chi_{483}(130,\cdot)\) \(\chi_{483}(172,\cdot)\) \(\chi_{483}(205,\cdot)\) \(\chi_{483}(214,\cdot)\) \(\chi_{483}(226,\cdot)\) \(\chi_{483}(235,\cdot)\) \(\chi_{483}(247,\cdot)\) \(\chi_{483}(268,\cdot)\) \(\chi_{483}(310,\cdot)\) \(\chi_{483}(319,\cdot)\) \(\chi_{483}(352,\cdot)\) \(\chi_{483}(373,\cdot)\) \(\chi_{483}(382,\cdot)\) \(\chi_{483}(424,\cdot)\) \(\chi_{483}(457,\cdot)\)
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
\((323,346,442)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 483 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) |
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)