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Properties

Label 997.193
Modulus $997$
Conductor $997$
Order $166$
Real no
Primitive yes
Minimal yes
Parity even

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(997, base_ring=CyclotomicField(166)) M = H._module chi = DirichletCharacter(H, M([7]))
 
Copy content gp:[g,chi] = znchar(Mod(193, 997))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("997.193");
 

Basic properties

Modulus: \(997\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(997\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(166\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 997.h

\(\chi_{997}(3,\cdot)\) \(\chi_{997}(4,\cdot)\) \(\chi_{997}(10,\cdot)\) \(\chi_{997}(25,\cdot)\) \(\chi_{997}(27,\cdot)\) \(\chi_{997}(31,\cdot)\) \(\chi_{997}(36,\cdot)\) \(\chi_{997}(48,\cdot)\) \(\chi_{997}(64,\cdot)\) \(\chi_{997}(83,\cdot)\) \(\chi_{997}(90,\cdot)\) \(\chi_{997}(97,\cdot)\) \(\chi_{997}(109,\cdot)\) \(\chi_{997}(120,\cdot)\) \(\chi_{997}(160,\cdot)\) \(\chi_{997}(167,\cdot)\) \(\chi_{997}(193,\cdot)\) \(\chi_{997}(199,\cdot)\) \(\chi_{997}(201,\cdot)\) \(\chi_{997}(222,\cdot)\) \(\chi_{997}(225,\cdot)\) \(\chi_{997}(243,\cdot)\) \(\chi_{997}(266,\cdot)\) \(\chi_{997}(268,\cdot)\) \(\chi_{997}(279,\cdot)\) \(\chi_{997}(296,\cdot)\) \(\chi_{997}(300,\cdot)\) \(\chi_{997}(311,\cdot)\) \(\chi_{997}(322,\cdot)\) \(\chi_{997}(324,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{7}{166}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 997 }(193, a) \) \(1\)\(1\)\(e\left(\frac{79}{166}\right)\)\(e\left(\frac{21}{83}\right)\)\(e\left(\frac{79}{83}\right)\)\(e\left(\frac{101}{166}\right)\)\(e\left(\frac{121}{166}\right)\)\(e\left(\frac{7}{166}\right)\)\(e\left(\frac{71}{166}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{7}{83}\right)\)\(e\left(\frac{75}{166}\right)\)
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)
 
\( \chi_{ 997 }(193,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 997 }(193,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 997 }(193,·),\chi_{ 997 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 997 }(193,·)) \;\) at \(\; a,b = \) e.g. 1,2