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Properties

Label 102.83
Modulus $102$
Conductor $51$
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(102, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,3]))
 
Copy content pari:[g,chi] = znchar(Mod(83,102))
 

Basic properties

Modulus: \(102\)
Conductor: \(51\)
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 \(\chi_{51}(32,\cdot)\)
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 102.g

\(\chi_{102}(53,\cdot)\) \(\chi_{102}(59,\cdot)\) \(\chi_{102}(77,\cdot)\) \(\chi_{102}(83,\cdot)\)

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: 8.0.33237432513.1

Values on generators

\((35,37)\) → \((-1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 102 }(83, a) \) \(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(i\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 102 }(83,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 102 }(83,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 102 }(83,·),\chi_{ 102 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 102 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2