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Properties

Label 64.25
Modulus $64$
Conductor $32$
Order $8$
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(64, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(25,64))
 

Basic properties

Modulus: \(64\)
Conductor: \(32\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{32}(5,\cdot)\)
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 64.g

\(\chi_{64}(9,\cdot)\) \(\chi_{64}(25,\cdot)\) \(\chi_{64}(41,\cdot)\) \(\chi_{64}(57,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{32})^+\)

Values on generators

\((63,5)\) → \((1,e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 64 }(25, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 64 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 64 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 64 }(25,·),\chi_{ 64 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 64 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2