Properties

Label 1024.257
Modulus $1024$
Conductor $16$
Order $4$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1024, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(257,1024))
 

Basic properties

Modulus: \(1024\)
Conductor: \(16\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{16}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1024.e

\(\chi_{1024}(257,\cdot)\) \(\chi_{1024}(769,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{16})^+\)

Values on generators

\((1023,5)\) → \((1,-i)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1024 }(257, a) \) \(1\)\(1\)\(i\)\(-i\)\(-1\)\(-1\)\(-i\)\(i\)\(1\)\(1\)\(i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1024 }(257,a) \;\) at \(\;a = \) e.g. 2