from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2448, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,2,0,5]))
pari: [g,chi] = znchar(Mod(1045,2448))
Basic properties
| Modulus: | \(2448\) | |
| Conductor: | \(272\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{272}(229,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2448.cj
\(\chi_{2448}(253,\cdot)\) \(\chi_{2448}(757,\cdot)\) \(\chi_{2448}(1045,\cdot)\) \(\chi_{2448}(1549,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.8.1721085137518592.2 |
Values on generators
\((2143,613,1361,1873)\) → \((1,i,1,e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2448 }(1045, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)