from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,48]))
pari: [g,chi] = znchar(Mod(3,1288))
Basic properties
| Modulus: | \(1288\) | |
| Conductor: | \(1288\) | 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 1288.cd
\(\chi_{1288}(3,\cdot)\) \(\chi_{1288}(59,\cdot)\) \(\chi_{1288}(75,\cdot)\) \(\chi_{1288}(131,\cdot)\) \(\chi_{1288}(187,\cdot)\) \(\chi_{1288}(243,\cdot)\) \(\chi_{1288}(395,\cdot)\) \(\chi_{1288}(579,\cdot)\) \(\chi_{1288}(675,\cdot)\) \(\chi_{1288}(731,\cdot)\) \(\chi_{1288}(859,\cdot)\) \(\chi_{1288}(899,\cdot)\) \(\chi_{1288}(915,\cdot)\) \(\chi_{1288}(955,\cdot)\) \(\chi_{1288}(1067,\cdot)\) \(\chi_{1288}(1083,\cdot)\) \(\chi_{1288}(1139,\cdot)\) \(\chi_{1288}(1179,\cdot)\) \(\chi_{1288}(1235,\cdot)\) \(\chi_{1288}(1251,\cdot)\)
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
\((967,645,185,281)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(3, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)