from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,8,3]))
pari: [g,chi] = znchar(Mod(1013,8619))
Basic properties
| Modulus: | \(8619\) | |
| Conductor: | \(663\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{663}(350,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8619.by
\(\chi_{8619}(1013,\cdot)\) \(\chi_{8619}(1520,\cdot)\) \(\chi_{8619}(3548,\cdot)\) \(\chi_{8619}(4562,\cdot)\) \(\chi_{8619}(5069,\cdot)\) \(\chi_{8619}(6083,\cdot)\) \(\chi_{8619}(6590,\cdot)\) \(\chi_{8619}(7604,\cdot)\)
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: | Number field defined by a degree 16 polynomial |
Values on generators
\((5747,5917,2536)\) → \((-1,-1,e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 8619 }(1013, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)