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Properties

Label 6021.347
Modulus $6021$
Conductor $6021$
Order $666$
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(6021, base_ring=CyclotomicField(666)) M = H._module chi = DirichletCharacter(H, M([407,660]))
 
Copy content gp:[g,chi] = znchar(Mod(347, 6021))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6021.347");
 

Basic properties

Modulus: \(6021\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6021\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(666\)
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 6021.cd

\(\chi_{6021}(29,\cdot)\) \(\chi_{6021}(47,\cdot)\) \(\chi_{6021}(50,\cdot)\) \(\chi_{6021}(65,\cdot)\) \(\chi_{6021}(74,\cdot)\) \(\chi_{6021}(83,\cdot)\) \(\chi_{6021}(86,\cdot)\) \(\chi_{6021}(131,\cdot)\) \(\chi_{6021}(146,\cdot)\) \(\chi_{6021}(218,\cdot)\) \(\chi_{6021}(248,\cdot)\) \(\chi_{6021}(266,\cdot)\) \(\chi_{6021}(281,\cdot)\) \(\chi_{6021}(317,\cdot)\) \(\chi_{6021}(347,\cdot)\) \(\chi_{6021}(353,\cdot)\) \(\chi_{6021}(356,\cdot)\) \(\chi_{6021}(371,\cdot)\) \(\chi_{6021}(389,\cdot)\) \(\chi_{6021}(401,\cdot)\) \(\chi_{6021}(515,\cdot)\) \(\chi_{6021}(527,\cdot)\) \(\chi_{6021}(608,\cdot)\) \(\chi_{6021}(659,\cdot)\) \(\chi_{6021}(707,\cdot)\) \(\chi_{6021}(722,\cdot)\) \(\chi_{6021}(770,\cdot)\) \(\chi_{6021}(779,\cdot)\) \(\chi_{6021}(785,\cdot)\) \(\chi_{6021}(812,\cdot)\) ...

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: $\Q(\zeta_{333})$
Fixed field: Number field defined by a degree 666 polynomial (not computed)

Values on generators

\((893,4240)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{110}{111}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6021 }(347, a) \) \(-1\)\(1\)\(e\left(\frac{659}{666}\right)\)\(e\left(\frac{326}{333}\right)\)\(e\left(\frac{169}{666}\right)\)\(e\left(\frac{295}{333}\right)\)\(e\left(\frac{215}{222}\right)\)\(e\left(\frac{9}{37}\right)\)\(e\left(\frac{653}{666}\right)\)\(e\left(\frac{188}{333}\right)\)\(e\left(\frac{583}{666}\right)\)\(e\left(\frac{319}{333}\right)\)
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)
 
\( \chi_{ 6021 }(347,a) \;\) at \(\;a = \) e.g. 2