from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2700, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(701,2700))
Basic properties
| Modulus: | \(2700\) | |
| Conductor: | \(3\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| Primitive: | no, induced from \(\chi_{3}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2700.g
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: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{-3}) \) |
Values on generators
\((1351,1001,2377)\) → \((1,-1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2700 }(701, a) \) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
sage: chi.jacobi_sum(n)