from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4080, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,8,0,11]))
pari: [g,chi] = znchar(Mod(41,4080))
Basic properties
| Modulus: | \(4080\) | |
| 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 \(\chi_{408}(245,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4080.ji
\(\chi_{4080}(41,\cdot)\) \(\chi_{4080}(521,\cdot)\) \(\chi_{4080}(1961,\cdot)\) \(\chi_{4080}(2441,\cdot)\) \(\chi_{4080}(2681,\cdot)\) \(\chi_{4080}(2921,\cdot)\) \(\chi_{4080}(3641,\cdot)\) \(\chi_{4080}(3881,\cdot)\)
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: | 16.16.315082116699567604562361581568.1 |
Values on generators
\((511,3061,1361,817,241)\) → \((1,-1,-1,1,e\left(\frac{11}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4080 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)