from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,8,8,7]))
pari: [g,chi] = znchar(Mod(827,1224))
Basic properties
| Modulus: | \(1224\) | |
| Conductor: | \(408\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{408}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1224.ce
\(\chi_{1224}(107,\cdot)\) \(\chi_{1224}(539,\cdot)\) \(\chi_{1224}(683,\cdot)\) \(\chi_{1224}(755,\cdot)\) \(\chi_{1224}(827,\cdot)\) \(\chi_{1224}(1043,\cdot)\) \(\chi_{1224}(1115,\cdot)\) \(\chi_{1224}(1187,\cdot)\)
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: | 16.0.315082116699567604562361581568.1 |
Values on generators
\((919,613,137,649)\) → \((-1,-1,-1,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1224 }(827, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)