from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([2,0,1]))
pari: [g,chi] = znchar(Mod(151,1300))
Basic properties
Modulus: | \(1300\) | |
Conductor: | \(52\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{52}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1300.j
\(\chi_{1300}(151,\cdot)\) \(\chi_{1300}(551,\cdot)\)
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: | \(\Q(\sqrt{-1}) \) |
Fixed field: | 4.4.35152.1 |
Values on generators
\((651,677,301)\) → \((-1,1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 1300 }(151, a) \) | \(1\) | \(1\) | \(-1\) | \(i\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) | \(-1\) | \(1\) |
sage: chi.jacobi_sum(n)