from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4034, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([15]))
pari: [g,chi] = znchar(Mod(691,4034))
Basic properties
| Modulus: | \(4034\) | |
| Conductor: | \(2017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2017}(691,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4034.k
\(\chi_{4034}(691,\cdot)\) \(\chi_{4034}(913,\cdot)\) \(\chi_{4034}(1489,\cdot)\) \(\chi_{4034}(1909,\cdot)\) \(\chi_{4034}(2125,\cdot)\) \(\chi_{4034}(2545,\cdot)\) \(\chi_{4034}(3121,\cdot)\) \(\chi_{4034}(3343,\cdot)\)
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
\(5\) → \(e\left(\frac{15}{16}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 4034 }(691, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)