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Properties

Label 780.131
Modulus $780$
Conductor $12$
Order $2$
Real yes
Primitive no
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(780, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1,1,0,0]))
 
Copy content pari:[g,chi] = znchar(Mod(131,780))
 

Basic properties

Modulus: \(780\)
Conductor: \(12\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{12}(11,\cdot)\)
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 780.g

\(\chi_{780}(131,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{3}) \)

Values on generators

\((391,521,157,301)\) → \((-1,-1,1,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 780 }(131, a) \) \(1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 780 }(131,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 780 }(131,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 780 }(131,·),\chi_{ 780 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 780 }(131,·)) \;\) at \(\; a,b = \) e.g. 1,2