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Properties

Label 1568.687
Modulus $1568$
Conductor $8$
Order $2$
Real yes
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(687,1568))
 

Basic properties

Modulus: \(1568\)
Conductor: \(8\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{8}(3,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1568.g

\(\chi_{1568}(687,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{-2}) \)

Values on generators

\((1471,197,1473)\) → \((-1,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1568 }(687, a) \) \(-1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1568 }(687,a) \;\) at \(\;a = \) e.g. 2