from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,27,55]))
pari: [g,chi] = znchar(Mod(296,4719))
Basic properties
| Modulus: | \(4719\) | |
| Conductor: | \(4719\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4719.cl
\(\chi_{4719}(296,\cdot)\) \(\chi_{4719}(329,\cdot)\) \(\chi_{4719}(758,\cdot)\) \(\chi_{4719}(1154,\cdot)\) \(\chi_{4719}(1187,\cdot)\) \(\chi_{4719}(1583,\cdot)\) \(\chi_{4719}(1616,\cdot)\) \(\chi_{4719}(2012,\cdot)\) \(\chi_{4719}(2045,\cdot)\) \(\chi_{4719}(2441,\cdot)\) \(\chi_{4719}(2474,\cdot)\) \(\chi_{4719}(2870,\cdot)\) \(\chi_{4719}(3299,\cdot)\) \(\chi_{4719}(3332,\cdot)\) \(\chi_{4719}(3728,\cdot)\) \(\chi_{4719}(3761,\cdot)\) \(\chi_{4719}(4157,\cdot)\) \(\chi_{4719}(4190,\cdot)\) \(\chi_{4719}(4586,\cdot)\) \(\chi_{4719}(4619,\cdot)\)
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
\((1574,3511,4357)\) → \((-1,e\left(\frac{9}{22}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4719 }(296, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) |
sage: chi.jacobi_sum(n)