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Properties

Label 771.170
Modulus $771$
Conductor $771$
Order $256$
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(771, base_ring=CyclotomicField(256)) M = H._module chi = DirichletCharacter(H, M([128,223]))
 
Copy content gp:[g,chi] = znchar(Mod(170, 771))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("771.170");
 

Basic properties

Modulus: \(771\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(771\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(256\)
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 771.q

\(\chi_{771}(5,\cdot)\) \(\chi_{771}(14,\cdot)\) \(\chi_{771}(20,\cdot)\) \(\chi_{771}(38,\cdot)\) \(\chi_{771}(41,\cdot)\) \(\chi_{771}(47,\cdot)\) \(\chi_{771}(53,\cdot)\) \(\chi_{771}(56,\cdot)\) \(\chi_{771}(65,\cdot)\) \(\chi_{771}(71,\cdot)\) \(\chi_{771}(74,\cdot)\) \(\chi_{771}(77,\cdot)\) \(\chi_{771}(80,\cdot)\) \(\chi_{771}(83,\cdot)\) \(\chi_{771}(86,\cdot)\) \(\chi_{771}(101,\cdot)\) \(\chi_{771}(107,\cdot)\) \(\chi_{771}(110,\cdot)\) \(\chi_{771}(119,\cdot)\) \(\chi_{771}(125,\cdot)\) \(\chi_{771}(131,\cdot)\) \(\chi_{771}(149,\cdot)\) \(\chi_{771}(152,\cdot)\) \(\chi_{771}(155,\cdot)\) \(\chi_{771}(161,\cdot)\) \(\chi_{771}(164,\cdot)\) \(\chi_{771}(167,\cdot)\) \(\chi_{771}(170,\cdot)\) \(\chi_{771}(179,\cdot)\) \(\chi_{771}(182,\cdot)\) ...

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: $\Q(\zeta_{256})$
Fixed field: Number field defined by a degree 256 polynomial (not computed)

Values on generators

\((515,517)\) → \((-1,e\left(\frac{223}{256}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 771 }(170, a) \) \(1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{105}{256}\right)\)\(e\left(\frac{11}{256}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{185}{256}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{43}{128}\right)\)\(e\left(\frac{91}{256}\right)\)\(i\)
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)
 
\( \chi_{ 771 }(170,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 771 }(170,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 771 }(170,·),\chi_{ 771 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 771 }(170,·)) \;\) at \(\; a,b = \) e.g. 1,2