from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,36,16]))
pari: [g,chi] = znchar(Mod(83,6080))
Basic properties
Modulus: | \(6080\) | |
Conductor: | \(6080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 6080.gp
\(\chi_{6080}(83,\cdot)\) \(\chi_{6080}(163,\cdot)\) \(\chi_{6080}(1147,\cdot)\) \(\chi_{6080}(1227,\cdot)\) \(\chi_{6080}(1603,\cdot)\) \(\chi_{6080}(1683,\cdot)\) \(\chi_{6080}(2667,\cdot)\) \(\chi_{6080}(2747,\cdot)\) \(\chi_{6080}(3123,\cdot)\) \(\chi_{6080}(3203,\cdot)\) \(\chi_{6080}(4187,\cdot)\) \(\chi_{6080}(4267,\cdot)\) \(\chi_{6080}(4643,\cdot)\) \(\chi_{6080}(4723,\cdot)\) \(\chi_{6080}(5707,\cdot)\) \(\chi_{6080}(5787,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((191,5701,1217,1921)\) → \((-1,e\left(\frac{7}{16}\right),-i,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 6080 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) |
sage: chi.jacobi_sum(n)