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Properties

Label 832.49
Modulus $832$
Conductor $208$
Order $12$
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(832, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,3,10]))
 
Copy content pari:[g,chi] = znchar(Mod(49,832))
 

Basic properties

Modulus: \(832\)
Conductor: \(208\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{208}(101,\cdot)\)
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 832.bp

\(\chi_{832}(17,\cdot)\) \(\chi_{832}(49,\cdot)\) \(\chi_{832}(433,\cdot)\) \(\chi_{832}(465,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.1184195427934675140608.1

Values on generators

\((703,261,769)\) → \((1,i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 832 }(49, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 832 }(49,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 832 }(49,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 832 }(49,·),\chi_{ 832 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 832 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2