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Properties

Label 2675.114
Modulus $2675$
Conductor $2675$
Order $530$
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(2675, base_ring=CyclotomicField(530)) M = H._module chi = DirichletCharacter(H, M([159,215]))
 
Copy content gp:[g,chi] = znchar(Mod(114, 2675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2675.114");
 

Basic properties

Modulus: \(2675\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2675\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(530\)
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 2675.v

\(\chi_{2675}(54,\cdot)\) \(\chi_{2675}(59,\cdot)\) \(\chi_{2675}(84,\cdot)\) \(\chi_{2675}(94,\cdot)\) \(\chi_{2675}(104,\cdot)\) \(\chi_{2675}(109,\cdot)\) \(\chi_{2675}(114,\cdot)\) \(\chi_{2675}(129,\cdot)\) \(\chi_{2675}(139,\cdot)\) \(\chi_{2675}(179,\cdot)\) \(\chi_{2675}(184,\cdot)\) \(\chi_{2675}(189,\cdot)\) \(\chi_{2675}(204,\cdot)\) \(\chi_{2675}(219,\cdot)\) \(\chi_{2675}(229,\cdot)\) \(\chi_{2675}(234,\cdot)\) \(\chi_{2675}(259,\cdot)\) \(\chi_{2675}(264,\cdot)\) \(\chi_{2675}(269,\cdot)\) \(\chi_{2675}(279,\cdot)\) \(\chi_{2675}(284,\cdot)\) \(\chi_{2675}(294,\cdot)\) \(\chi_{2675}(309,\cdot)\) \(\chi_{2675}(329,\cdot)\) \(\chi_{2675}(339,\cdot)\) \(\chi_{2675}(359,\cdot)\) \(\chi_{2675}(364,\cdot)\) \(\chi_{2675}(379,\cdot)\) \(\chi_{2675}(384,\cdot)\) \(\chi_{2675}(389,\cdot)\) ...

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: $\Q(\zeta_{265})$
Fixed field: Number field defined by a degree 530 polynomial (not computed)

Values on generators

\((1927,751)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{43}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2675 }(114, a) \) \(-1\)\(1\)\(e\left(\frac{187}{265}\right)\)\(e\left(\frac{263}{530}\right)\)\(e\left(\frac{109}{265}\right)\)\(e\left(\frac{107}{530}\right)\)\(e\left(\frac{50}{53}\right)\)\(e\left(\frac{31}{265}\right)\)\(e\left(\frac{263}{265}\right)\)\(e\left(\frac{192}{265}\right)\)\(e\left(\frac{481}{530}\right)\)\(e\left(\frac{201}{530}\right)\)
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)
 
\( \chi_{ 2675 }(114,a) \;\) at \(\;a = \) e.g. 2