from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13,8,4]))
pari: [g,chi] = znchar(Mod(107,4800))
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{960}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.cx
\(\chi_{4800}(107,\cdot)\) \(\chi_{4800}(1043,\cdot)\) \(\chi_{4800}(1307,\cdot)\) \(\chi_{4800}(2243,\cdot)\) \(\chi_{4800}(2507,\cdot)\) \(\chi_{4800}(3443,\cdot)\) \(\chi_{4800}(3707,\cdot)\) \(\chi_{4800}(4643,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.0.968232702940866945220608000000000000.2 |
Values on generators
\((4351,901,1601,577)\) → \((-1,e\left(\frac{13}{16}\right),-1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4800 }(107, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)