from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,1,8,0]))
pari: [g,chi] = znchar(Mod(251,4800))
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(192\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{192}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4800.dd
\(\chi_{4800}(251,\cdot)\) \(\chi_{4800}(851,\cdot)\) \(\chi_{4800}(1451,\cdot)\) \(\chi_{4800}(2051,\cdot)\) \(\chi_{4800}(2651,\cdot)\) \(\chi_{4800}(3251,\cdot)\) \(\chi_{4800}(3851,\cdot)\) \(\chi_{4800}(4451,\cdot)\)
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: | 16.16.3965881151245791007623610368.1 |
Values on generators
\((4351,901,1601,577)\) → \((-1,e\left(\frac{1}{16}\right),-1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4800 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)