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Properties

Label 1065.113
Modulus $1065$
Conductor $1065$
Order $140$
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(1065, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([70,105,66]))
 
Copy content gp:[g,chi] = znchar(Mod(113, 1065))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1065.113");
 

Basic properties

Modulus: \(1065\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1065\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(140\)
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 1065.bv

\(\chi_{1065}(47,\cdot)\) \(\chi_{1065}(53,\cdot)\) \(\chi_{1065}(62,\cdot)\) \(\chi_{1065}(68,\cdot)\) \(\chi_{1065}(92,\cdot)\) \(\chi_{1065}(113,\cdot)\) \(\chi_{1065}(173,\cdot)\) \(\chi_{1065}(197,\cdot)\) \(\chi_{1065}(203,\cdot)\) \(\chi_{1065}(248,\cdot)\) \(\chi_{1065}(257,\cdot)\) \(\chi_{1065}(272,\cdot)\) \(\chi_{1065}(278,\cdot)\) \(\chi_{1065}(317,\cdot)\) \(\chi_{1065}(347,\cdot)\) \(\chi_{1065}(353,\cdot)\) \(\chi_{1065}(362,\cdot)\) \(\chi_{1065}(368,\cdot)\) \(\chi_{1065}(377,\cdot)\) \(\chi_{1065}(383,\cdot)\) \(\chi_{1065}(407,\cdot)\) \(\chi_{1065}(422,\cdot)\) \(\chi_{1065}(437,\cdot)\) \(\chi_{1065}(473,\cdot)\) \(\chi_{1065}(482,\cdot)\) \(\chi_{1065}(488,\cdot)\) \(\chi_{1065}(518,\cdot)\) \(\chi_{1065}(623,\cdot)\) \(\chi_{1065}(683,\cdot)\) \(\chi_{1065}(692,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((356,427,646)\) → \((-1,-i,e\left(\frac{33}{70}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1065 }(113, a) \) \(-1\)\(1\)\(e\left(\frac{11}{140}\right)\)\(e\left(\frac{11}{70}\right)\)\(e\left(\frac{31}{140}\right)\)\(e\left(\frac{33}{140}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{89}{140}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{70}\right)\)
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)
 
\( \chi_{ 1065 }(113,a) \;\) at \(\;a = \) e.g. 2