from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,11,0,12]))
pari: [g,chi] = znchar(Mod(163,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{320}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.eh
\(\chi_{8640}(163,\cdot)\) \(\chi_{8640}(2107,\cdot)\) \(\chi_{8640}(2323,\cdot)\) \(\chi_{8640}(4267,\cdot)\) \(\chi_{8640}(4483,\cdot)\) \(\chi_{8640}(6427,\cdot)\) \(\chi_{8640}(6643,\cdot)\) \(\chi_{8640}(8587,\cdot)\)
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: | 16.16.147573952589676412928000000000000.2 |
Values on generators
\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{11}{16}\right),1,-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(163, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)