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Properties

Label 608.419
Modulus $608$
Conductor $32$
Order $8$
Real no
Primitive no
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(608, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,3,0]))
 
Copy content pari:[g,chi] = znchar(Mod(419,608))
 

Basic properties

Modulus: \(608\)
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 \(\chi_{32}(3,\cdot)\)
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 608.x

\(\chi_{608}(115,\cdot)\) \(\chi_{608}(267,\cdot)\) \(\chi_{608}(419,\cdot)\) \(\chi_{608}(571,\cdot)\)

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: 8.0.2147483648.1

Values on generators

\((191,229,97)\) → \((-1,e\left(\frac{3}{8}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 608 }(419, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 608 }(419,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 608 }(419,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 608 }(419,·),\chi_{ 608 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 608 }(419,·)) \;\) at \(\; a,b = \) e.g. 1,2