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Properties

Label 1135.13
Modulus $1135$
Conductor $1135$
Order $452$
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(1135, base_ring=CyclotomicField(452)) M = H._module chi = DirichletCharacter(H, M([339,122]))
 
Copy content gp:[g,chi] = znchar(Mod(13, 1135))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1135.13");
 

Basic properties

Modulus: \(1135\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1135\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(452\)
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 1135.l

\(\chi_{1135}(2,\cdot)\) \(\chi_{1135}(8,\cdot)\) \(\chi_{1135}(13,\cdot)\) \(\chi_{1135}(17,\cdot)\) \(\chi_{1135}(18,\cdot)\) \(\chi_{1135}(22,\cdot)\) \(\chi_{1135}(32,\cdot)\) \(\chi_{1135}(37,\cdot)\) \(\chi_{1135}(38,\cdot)\) \(\chi_{1135}(42,\cdot)\) \(\chi_{1135}(52,\cdot)\) \(\chi_{1135}(58,\cdot)\) \(\chi_{1135}(67,\cdot)\) \(\chi_{1135}(68,\cdot)\) \(\chi_{1135}(72,\cdot)\) \(\chi_{1135}(83,\cdot)\) \(\chi_{1135}(88,\cdot)\) \(\chi_{1135}(93,\cdot)\) \(\chi_{1135}(98,\cdot)\) \(\chi_{1135}(107,\cdot)\) \(\chi_{1135}(117,\cdot)\) \(\chi_{1135}(118,\cdot)\) \(\chi_{1135}(123,\cdot)\) \(\chi_{1135}(127,\cdot)\) \(\chi_{1135}(128,\cdot)\) \(\chi_{1135}(137,\cdot)\) \(\chi_{1135}(138,\cdot)\) \(\chi_{1135}(142,\cdot)\) \(\chi_{1135}(143,\cdot)\) \(\chi_{1135}(148,\cdot)\) ...

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: $\Q(\zeta_{452})$
Fixed field: Number field defined by a degree 452 polynomial (not computed)

Values on generators

\((682,456)\) → \((-i,e\left(\frac{61}{226}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1135 }(13, a) \) \(1\)\(1\)\(e\left(\frac{9}{452}\right)\)\(e\left(\frac{301}{452}\right)\)\(e\left(\frac{9}{226}\right)\)\(e\left(\frac{155}{226}\right)\)\(e\left(\frac{143}{452}\right)\)\(e\left(\frac{27}{452}\right)\)\(e\left(\frac{75}{226}\right)\)\(e\left(\frac{63}{113}\right)\)\(e\left(\frac{319}{452}\right)\)\(e\left(\frac{323}{452}\right)\)
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)
 
\( \chi_{ 1135 }(13,a) \;\) at \(\;a = \) e.g. 2