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Properties

Label 656.163
Modulus $656$
Conductor $656$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(656, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(163,656))
 

Basic properties

Modulus: \(656\)
Conductor: \(656\)
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: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 656.m

\(\chi_{656}(163,\cdot)\) \(\chi_{656}(491,\cdot)\)

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: 4.0.3442688.5

Values on generators

\((575,165,129)\) → \((-1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 656 }(163, a) \) \(-1\)\(1\)\(i\)\(-i\)\(-1\)\(-1\)\(-i\)\(-i\)\(1\)\(-1\)\(i\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 656 }(163,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 656 }(163,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 656 }(163,·),\chi_{ 656 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 656 }(163,·)) \;\) at \(\; a,b = \) e.g. 1,2