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Properties

Label 128.109
Modulus $128$
Conductor $128$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(32)) M = H._module chi = DirichletCharacter(H, M([0,23]))
 
Copy content pari:[g,chi] = znchar(Mod(109,128))
 

Basic properties

Modulus: \(128\)
Conductor: \(128\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(32\)
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: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 128.k

\(\chi_{128}(5,\cdot)\) \(\chi_{128}(13,\cdot)\) \(\chi_{128}(21,\cdot)\) \(\chi_{128}(29,\cdot)\) \(\chi_{128}(37,\cdot)\) \(\chi_{128}(45,\cdot)\) \(\chi_{128}(53,\cdot)\) \(\chi_{128}(61,\cdot)\) \(\chi_{128}(69,\cdot)\) \(\chi_{128}(77,\cdot)\) \(\chi_{128}(85,\cdot)\) \(\chi_{128}(93,\cdot)\) \(\chi_{128}(101,\cdot)\) \(\chi_{128}(109,\cdot)\) \(\chi_{128}(117,\cdot)\) \(\chi_{128}(125,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: \(\Q(\zeta_{128})^+\)

Values on generators

\((127,5)\) → \((1,e\left(\frac{23}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 128 }(109, a) \) \(1\)\(1\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{11}{32}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 128 }(109,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 128 }(109,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 128 }(109,·),\chi_{ 128 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 128 }(109,·)) \;\) at \(\; a,b = \) e.g. 1,2