from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,44,12]))
pari: [g,chi] = znchar(Mod(165,1288))
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.by
\(\chi_{1288}(165,\cdot)\) \(\chi_{1288}(261,\cdot)\) \(\chi_{1288}(317,\cdot)\) \(\chi_{1288}(445,\cdot)\) \(\chi_{1288}(485,\cdot)\) \(\chi_{1288}(501,\cdot)\) \(\chi_{1288}(541,\cdot)\) \(\chi_{1288}(653,\cdot)\) \(\chi_{1288}(669,\cdot)\) \(\chi_{1288}(725,\cdot)\) \(\chi_{1288}(765,\cdot)\) \(\chi_{1288}(821,\cdot)\) \(\chi_{1288}(837,\cdot)\) \(\chi_{1288}(877,\cdot)\) \(\chi_{1288}(933,\cdot)\) \(\chi_{1288}(949,\cdot)\) \(\chi_{1288}(1005,\cdot)\) \(\chi_{1288}(1061,\cdot)\) \(\chi_{1288}(1117,\cdot)\) \(\chi_{1288}(1269,\cdot)\)
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
\((967,645,185,281)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 1288 }(165, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)