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Properties

Label 2675.13
Modulus $2675$
Conductor $2675$
Order $1060$
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(1060)) M = H._module chi = DirichletCharacter(H, M([1007,140]))
 
Copy content gp:[g,chi] = znchar(Mod(13, 2675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2675.13");
 

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: 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.x

\(\chi_{2675}(3,\cdot)\) \(\chi_{2675}(12,\cdot)\) \(\chi_{2675}(13,\cdot)\) \(\chi_{2675}(23,\cdot)\) \(\chi_{2675}(27,\cdot)\) \(\chi_{2675}(33,\cdot)\) \(\chi_{2675}(37,\cdot)\) \(\chi_{2675}(42,\cdot)\) \(\chi_{2675}(47,\cdot)\) \(\chi_{2675}(48,\cdot)\) \(\chi_{2675}(52,\cdot)\) \(\chi_{2675}(53,\cdot)\) \(\chi_{2675}(62,\cdot)\) \(\chi_{2675}(83,\cdot)\) \(\chi_{2675}(87,\cdot)\) \(\chi_{2675}(92,\cdot)\) \(\chi_{2675}(102,\cdot)\) \(\chi_{2675}(117,\cdot)\) \(\chi_{2675}(123,\cdot)\) \(\chi_{2675}(137,\cdot)\) \(\chi_{2675}(142,\cdot)\) \(\chi_{2675}(147,\cdot)\) \(\chi_{2675}(148,\cdot)\) \(\chi_{2675}(163,\cdot)\) \(\chi_{2675}(183,\cdot)\) \(\chi_{2675}(188,\cdot)\) \(\chi_{2675}(192,\cdot)\) \(\chi_{2675}(197,\cdot)\) \(\chi_{2675}(208,\cdot)\) \(\chi_{2675}(212,\cdot)\) ...

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

\((1927,751)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{7}{53}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2675 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{87}{1060}\right)\)\(e\left(\frac{949}{1060}\right)\)\(e\left(\frac{87}{530}\right)\)\(e\left(\frac{259}{265}\right)\)\(e\left(\frac{91}{212}\right)\)\(e\left(\frac{261}{1060}\right)\)\(e\left(\frac{419}{530}\right)\)\(e\left(\frac{28}{265}\right)\)\(e\left(\frac{63}{1060}\right)\)\(e\left(\frac{953}{1060}\right)\)
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)
 
\( \chi_{ 2675 }(13,a) \;\) at \(\;a = \) e.g. 2