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Properties

Label 967.63
Modulus $967$
Conductor $967$
Order $966$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(967, base_ring=CyclotomicField(966)) M = H._module chi = DirichletCharacter(H, M([541]))
 
Copy content gp:[g,chi] = znchar(Mod(63, 967))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("967.63");
 

Basic properties

Modulus: \(967\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(967\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(966\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 967.p

\(\chi_{967}(5,\cdot)\) \(\chi_{967}(6,\cdot)\) \(\chi_{967}(7,\cdot)\) \(\chi_{967}(12,\cdot)\) \(\chi_{967}(13,\cdot)\) \(\chi_{967}(19,\cdot)\) \(\chi_{967}(28,\cdot)\) \(\chi_{967}(37,\cdot)\) \(\chi_{967}(38,\cdot)\) \(\chi_{967}(40,\cdot)\) \(\chi_{967}(43,\cdot)\) \(\chi_{967}(45,\cdot)\) \(\chi_{967}(46,\cdot)\) \(\chi_{967}(47,\cdot)\) \(\chi_{967}(48,\cdot)\) \(\chi_{967}(56,\cdot)\) \(\chi_{967}(58,\cdot)\) \(\chi_{967}(63,\cdot)\) \(\chi_{967}(66,\cdot)\) \(\chi_{967}(75,\cdot)\) \(\chi_{967}(77,\cdot)\) \(\chi_{967}(79,\cdot)\) \(\chi_{967}(82,\cdot)\) \(\chi_{967}(85,\cdot)\) \(\chi_{967}(86,\cdot)\) \(\chi_{967}(89,\cdot)\) \(\chi_{967}(102,\cdot)\) \(\chi_{967}(104,\cdot)\) \(\chi_{967}(105,\cdot)\) \(\chi_{967}(107,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{541}{966}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 967 }(63, a) \) \(-1\)\(1\)\(e\left(\frac{220}{483}\right)\)\(e\left(\frac{235}{322}\right)\)\(e\left(\frac{440}{483}\right)\)\(e\left(\frac{541}{966}\right)\)\(e\left(\frac{179}{966}\right)\)\(e\left(\frac{505}{966}\right)\)\(e\left(\frac{59}{161}\right)\)\(e\left(\frac{74}{161}\right)\)\(e\left(\frac{5}{322}\right)\)\(e\left(\frac{51}{161}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 967 }(63,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(63,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(63,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(63,·)) \;\) at \(\; a,b = \) e.g. 1,2