from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,15,0]))
pari: [g,chi] = znchar(Mod(1135,2304))
Basic properties
| Modulus: | \(2304\) | |
| Conductor: | \(64\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{64}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2304.bf
\(\chi_{2304}(271,\cdot)\) \(\chi_{2304}(559,\cdot)\) \(\chi_{2304}(847,\cdot)\) \(\chi_{2304}(1135,\cdot)\) \(\chi_{2304}(1423,\cdot)\) \(\chi_{2304}(1711,\cdot)\) \(\chi_{2304}(1999,\cdot)\) \(\chi_{2304}(2287,\cdot)\)
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: | 16.0.604462909807314587353088.1 |
Values on generators
\((1279,2053,1793)\) → \((-1,e\left(\frac{15}{16}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2304 }(1135, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)