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Properties

Label 2675.113
Modulus $2675$
Conductor $2675$
Order $1060$
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(2675, base_ring=CyclotomicField(1060)) M = H._module chi = DirichletCharacter(H, M([1007,710]))
 
Copy content gp:[g,chi] = znchar(Mod(113, 2675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2675.113");
 

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: \(1060\)
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 2675.w

\(\chi_{2675}(2,\cdot)\) \(\chi_{2675}(8,\cdot)\) \(\chi_{2675}(17,\cdot)\) \(\chi_{2675}(22,\cdot)\) \(\chi_{2675}(28,\cdot)\) \(\chi_{2675}(38,\cdot)\) \(\chi_{2675}(58,\cdot)\) \(\chi_{2675}(63,\cdot)\) \(\chi_{2675}(67,\cdot)\) \(\chi_{2675}(72,\cdot)\) \(\chi_{2675}(73,\cdot)\) \(\chi_{2675}(77,\cdot)\) \(\chi_{2675}(78,\cdot)\) \(\chi_{2675}(88,\cdot)\) \(\chi_{2675}(97,\cdot)\) \(\chi_{2675}(98,\cdot)\) \(\chi_{2675}(103,\cdot)\) \(\chi_{2675}(112,\cdot)\) \(\chi_{2675}(113,\cdot)\) \(\chi_{2675}(122,\cdot)\) \(\chi_{2675}(127,\cdot)\) \(\chi_{2675}(128,\cdot)\) \(\chi_{2675}(133,\cdot)\) \(\chi_{2675}(138,\cdot)\) \(\chi_{2675}(152,\cdot)\) \(\chi_{2675}(153,\cdot)\) \(\chi_{2675}(158,\cdot)\) \(\chi_{2675}(162,\cdot)\) \(\chi_{2675}(167,\cdot)\) \(\chi_{2675}(172,\cdot)\) ...

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: $\Q(\zeta_{1060})$
Fixed field: Number field defined by a degree 1060 polynomial (not computed)

Values on generators

\((1927,751)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{71}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2675 }(113, a) \) \(1\)\(1\)\(e\left(\frac{657}{1060}\right)\)\(e\left(\frac{569}{1060}\right)\)\(e\left(\frac{127}{530}\right)\)\(e\left(\frac{83}{530}\right)\)\(e\left(\frac{117}{212}\right)\)\(e\left(\frac{911}{1060}\right)\)\(e\left(\frac{39}{530}\right)\)\(e\left(\frac{248}{265}\right)\)\(e\left(\frac{823}{1060}\right)\)\(e\left(\frac{453}{1060}\right)\)
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)
 
\( \chi_{ 2675 }(113,a) \;\) at \(\;a = \) e.g. 2