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Properties

Label 1280.1023
Modulus $1280$
Conductor $20$
Order $4$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,0,3]))
 
Copy content pari:[g,chi] = znchar(Mod(1023,1280))
 

Basic properties

Modulus: \(1280\)
Conductor: \(20\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{20}(3,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1280.n

\(\chi_{1280}(767,\cdot)\) \(\chi_{1280}(1023,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari: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: \(\Q(\zeta_{20})^+\)

Values on generators

\((511,261,257)\) → \((-1,1,-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1280 }(1023, a) \) \(1\)\(1\)\(-i\)\(i\)\(-1\)\(-1\)\(i\)\(-i\)\(1\)\(1\)\(-i\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1280 }(1023,a) \;\) at \(\;a = \) e.g. 2