from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,11,18]))
pari: [g,chi] = znchar(Mod(31,1288))
Basic properties
| Modulus: | \(1288\) | |
| Conductor: | \(644\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{644}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1288.cg
\(\chi_{1288}(31,\cdot)\) \(\chi_{1288}(87,\cdot)\) \(\chi_{1288}(215,\cdot)\) \(\chi_{1288}(255,\cdot)\) \(\chi_{1288}(271,\cdot)\) \(\chi_{1288}(311,\cdot)\) \(\chi_{1288}(423,\cdot)\) \(\chi_{1288}(439,\cdot)\) \(\chi_{1288}(495,\cdot)\) \(\chi_{1288}(535,\cdot)\) \(\chi_{1288}(591,\cdot)\) \(\chi_{1288}(607,\cdot)\) \(\chi_{1288}(647,\cdot)\) \(\chi_{1288}(703,\cdot)\) \(\chi_{1288}(719,\cdot)\) \(\chi_{1288}(775,\cdot)\) \(\chi_{1288}(831,\cdot)\) \(\chi_{1288}(887,\cdot)\) \(\chi_{1288}(1039,\cdot)\) \(\chi_{1288}(1223,\cdot)\)
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
\((967,645,185,281)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) |
sage: chi.jacobi_sum(n)