from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,30]))
pari: [g,chi] = znchar(Mod(1244,4235))
Basic properties
Modulus: | \(4235\) | |
Conductor: | \(4235\) | 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 4235.ck
\(\chi_{4235}(89,\cdot)\) \(\chi_{4235}(199,\cdot)\) \(\chi_{4235}(474,\cdot)\) \(\chi_{4235}(584,\cdot)\) \(\chi_{4235}(859,\cdot)\) \(\chi_{4235}(1244,\cdot)\) \(\chi_{4235}(1354,\cdot)\) \(\chi_{4235}(1629,\cdot)\) \(\chi_{4235}(1739,\cdot)\) \(\chi_{4235}(2014,\cdot)\) \(\chi_{4235}(2124,\cdot)\) \(\chi_{4235}(2399,\cdot)\) \(\chi_{4235}(2509,\cdot)\) \(\chi_{4235}(2894,\cdot)\) \(\chi_{4235}(3169,\cdot)\) \(\chi_{4235}(3279,\cdot)\) \(\chi_{4235}(3554,\cdot)\) \(\chi_{4235}(3664,\cdot)\) \(\chi_{4235}(3939,\cdot)\) \(\chi_{4235}(4049,\cdot)\)
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
\((2542,1816,2906)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 4235 }(1244, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) |
sage: chi.jacobi_sum(n)