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Properties

Label 1002.271
Modulus $1002$
Conductor $167$
Order $166$
Real no
Primitive no
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(1002, base_ring=CyclotomicField(166)) M = H._module chi = DirichletCharacter(H, M([0,57]))
 
Copy content gp:[g,chi] = znchar(Mod(271, 1002))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1002.271");
 

Basic properties

Modulus: \(1002\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(167\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(166\)
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: no, induced from \(\chi_{167}(104,\cdot)\)
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 1002.g

\(\chi_{1002}(13,\cdot)\) \(\chi_{1002}(37,\cdot)\) \(\chi_{1002}(43,\cdot)\) \(\chi_{1002}(55,\cdot)\) \(\chi_{1002}(67,\cdot)\) \(\chi_{1002}(73,\cdot)\) \(\chi_{1002}(79,\cdot)\) \(\chi_{1002}(91,\cdot)\) \(\chi_{1002}(103,\cdot)\) \(\chi_{1002}(109,\cdot)\) \(\chi_{1002}(139,\cdot)\) \(\chi_{1002}(145,\cdot)\) \(\chi_{1002}(151,\cdot)\) \(\chi_{1002}(163,\cdot)\) \(\chi_{1002}(187,\cdot)\) \(\chi_{1002}(193,\cdot)\) \(\chi_{1002}(235,\cdot)\) \(\chi_{1002}(241,\cdot)\) \(\chi_{1002}(247,\cdot)\) \(\chi_{1002}(253,\cdot)\) \(\chi_{1002}(259,\cdot)\) \(\chi_{1002}(271,\cdot)\) \(\chi_{1002}(277,\cdot)\) \(\chi_{1002}(301,\cdot)\) \(\chi_{1002}(307,\cdot)\) \(\chi_{1002}(313,\cdot)\) \(\chi_{1002}(325,\cdot)\) \(\chi_{1002}(331,\cdot)\) \(\chi_{1002}(349,\cdot)\) \(\chi_{1002}(373,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((335,673)\) → \((1,e\left(\frac{57}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1002 }(271, a) \) \(-1\)\(1\)\(e\left(\frac{57}{166}\right)\)\(e\left(\frac{43}{83}\right)\)\(e\left(\frac{51}{83}\right)\)\(e\left(\frac{61}{166}\right)\)\(e\left(\frac{33}{166}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{165}{166}\right)\)\(e\left(\frac{57}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{75}{83}\right)\)
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)
 
\( \chi_{ 1002 }(271,a) \;\) at \(\;a = \) e.g. 2