from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,13,8,4]))
pari: [g,chi] = znchar(Mod(917,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{960}(917,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.du
\(\chi_{8640}(917,\cdot)\) \(\chi_{8640}(1133,\cdot)\) \(\chi_{8640}(3077,\cdot)\) \(\chi_{8640}(3293,\cdot)\) \(\chi_{8640}(5237,\cdot)\) \(\chi_{8640}(5453,\cdot)\) \(\chi_{8640}(7397,\cdot)\) \(\chi_{8640}(7613,\cdot)\)
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: | 16.16.968232702940866945220608000000000000.2 |
Values on generators
\((2431,3781,6401,3457)\) → \((1,e\left(\frac{13}{16}\right),-1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(917, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)