from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,24,8]))
pari: [g,chi] = znchar(Mod(179,6080))
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.gt
\(\chi_{6080}(179,\cdot)\) \(\chi_{6080}(259,\cdot)\) \(\chi_{6080}(939,\cdot)\) \(\chi_{6080}(1019,\cdot)\) \(\chi_{6080}(1699,\cdot)\) \(\chi_{6080}(1779,\cdot)\) \(\chi_{6080}(2459,\cdot)\) \(\chi_{6080}(2539,\cdot)\) \(\chi_{6080}(3219,\cdot)\) \(\chi_{6080}(3299,\cdot)\) \(\chi_{6080}(3979,\cdot)\) \(\chi_{6080}(4059,\cdot)\) \(\chi_{6080}(4739,\cdot)\) \(\chi_{6080}(4819,\cdot)\) \(\chi_{6080}(5499,\cdot)\) \(\chi_{6080}(5579,\cdot)\)
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
\((191,5701,1217,1921)\) → \((-1,e\left(\frac{15}{16}\right),-1,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6080 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) |
sage: chi.jacobi_sum(n)