from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,6,16]))
pari: [g,chi] = znchar(Mod(87,6080))
Basic properties
Modulus: | \(6080\) | |
Conductor: | \(3040\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3040}(467,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6080.ey
\(\chi_{6080}(87,\cdot)\) \(\chi_{6080}(1223,\cdot)\) \(\chi_{6080}(1527,\cdot)\) \(\chi_{6080}(2823,\cdot)\) \(\chi_{6080}(3127,\cdot)\) \(\chi_{6080}(4263,\cdot)\) \(\chi_{6080}(4567,\cdot)\) \(\chi_{6080}(5863,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((191,5701,1217,1921)\) → \((-1,e\left(\frac{7}{8}\right),i,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 6080 }(87, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) |
sage: chi.jacobi_sum(n)