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Properties

Label 3151.1010
Modulus $3151$
Conductor $3151$
Order $1496$
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(3151, base_ring=CyclotomicField(1496)) M = H._module chi = DirichletCharacter(H, M([884,429]))
 
Copy content gp:[g,chi] = znchar(Mod(1010, 3151))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3151.1010");
 

Basic properties

Modulus: \(3151\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3151\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1496\)
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 3151.bf

\(\chi_{3151}(5,\cdot)\) \(\chi_{3151}(20,\cdot)\) \(\chi_{3151}(21,\cdot)\) \(\chi_{3151}(33,\cdot)\) \(\chi_{3151}(40,\cdot)\) \(\chi_{3151}(42,\cdot)\) \(\chi_{3151}(43,\cdot)\) \(\chi_{3151}(51,\cdot)\) \(\chi_{3151}(53,\cdot)\) \(\chi_{3151}(57,\cdot)\) \(\chi_{3151}(66,\cdot)\) \(\chi_{3151}(67,\cdot)\) \(\chi_{3151}(79,\cdot)\) \(\chi_{3151}(80,\cdot)\) \(\chi_{3151}(83,\cdot)\) \(\chi_{3151}(84,\cdot)\) \(\chi_{3151}(86,\cdot)\) \(\chi_{3151}(89,\cdot)\) \(\chi_{3151}(90,\cdot)\) \(\chi_{3151}(97,\cdot)\) \(\chi_{3151}(102,\cdot)\) \(\chi_{3151}(106,\cdot)\) \(\chi_{3151}(111,\cdot)\) \(\chi_{3151}(113,\cdot)\) \(\chi_{3151}(125,\cdot)\) \(\chi_{3151}(132,\cdot)\) \(\chi_{3151}(134,\cdot)\) \(\chi_{3151}(143,\cdot)\) \(\chi_{3151}(149,\cdot)\) \(\chi_{3151}(157,\cdot)\) ...

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: $\Q(\zeta_{1496})$
Fixed field: Number field defined by a degree 1496 polynomial (not computed)

Values on generators

\((2604,277)\) → \((e\left(\frac{13}{22}\right),e\left(\frac{39}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3151 }(1010, a) \) \(1\)\(1\)\(e\left(\frac{37}{748}\right)\)\(e\left(\frac{1109}{1496}\right)\)\(e\left(\frac{37}{374}\right)\)\(e\left(\frac{147}{1496}\right)\)\(e\left(\frac{1183}{1496}\right)\)\(e\left(\frac{203}{748}\right)\)\(e\left(\frac{111}{748}\right)\)\(e\left(\frac{361}{748}\right)\)\(e\left(\frac{13}{88}\right)\)\(e\left(\frac{227}{748}\right)\)
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)
 
\( \chi_{ 3151 }(1010,a) \;\) at \(\;a = \) e.g. 2