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Properties

Label 3076.7
Modulus $3076$
Conductor $3076$
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(3076, base_ring=CyclotomicField(256)) M = H._module chi = DirichletCharacter(H, M([128,153]))
 
Copy content gp:[g,chi] = znchar(Mod(7, 3076))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3076.7");
 

Basic properties

Modulus: \(3076\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3076\)
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 3076.be

\(\chi_{3076}(7,\cdot)\) \(\chi_{3076}(35,\cdot)\) \(\chi_{3076}(39,\cdot)\) \(\chi_{3076}(67,\cdot)\) \(\chi_{3076}(79,\cdot)\) \(\chi_{3076}(111,\cdot)\) \(\chi_{3076}(119,\cdot)\) \(\chi_{3076}(139,\cdot)\) \(\chi_{3076}(159,\cdot)\) \(\chi_{3076}(175,\cdot)\) \(\chi_{3076}(179,\cdot)\) \(\chi_{3076}(183,\cdot)\) \(\chi_{3076}(195,\cdot)\) \(\chi_{3076}(207,\cdot)\) \(\chi_{3076}(219,\cdot)\) \(\chi_{3076}(239,\cdot)\) \(\chi_{3076}(271,\cdot)\) \(\chi_{3076}(327,\cdot)\) \(\chi_{3076}(335,\cdot)\) \(\chi_{3076}(343,\cdot)\) \(\chi_{3076}(395,\cdot)\) \(\chi_{3076}(399,\cdot)\) \(\chi_{3076}(443,\cdot)\) \(\chi_{3076}(503,\cdot)\) \(\chi_{3076}(551,\cdot)\) \(\chi_{3076}(555,\cdot)\) \(\chi_{3076}(563,\cdot)\) \(\chi_{3076}(595,\cdot)\) \(\chi_{3076}(623,\cdot)\) \(\chi_{3076}(631,\cdot)\) ...

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

\((1539,1549)\) → \((-1,e\left(\frac{153}{256}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3076 }(7, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{119}{128}\right)\)\(e\left(\frac{211}{256}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{25}{256}\right)\)\(e\left(\frac{15}{256}\right)\)\(e\left(\frac{95}{128}\right)\)\(e\left(\frac{31}{128}\right)\)\(i\)\(e\left(\frac{163}{256}\right)\)
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)
 
\( \chi_{ 3076 }(7,a) \;\) at \(\;a = \) e.g. 2