from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2020, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,3,2]))
pari: [g,chi] = znchar(Mod(1413,2020))
Basic properties
| Modulus: | \(2020\) | |
| Conductor: | \(505\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{505}(403,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2020.k
\(\chi_{2020}(1413,\cdot)\) \(\chi_{2020}(1817,\cdot)\)
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: | \(\mathbb{Q}(i)\) |
| Fixed field: | 4.0.1275125.2 |
Values on generators
\((1011,1617,1921)\) → \((1,-i,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2020 }(1413, a) \) | \(-1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(-1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(i\) |
sage: chi.jacobi_sum(n)