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Properties

Label 2021.137
Modulus $2021$
Conductor $2021$
Order $322$
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(2021, base_ring=CyclotomicField(322)) M = H._module chi = DirichletCharacter(H, M([299,91]))
 
Copy content gp:[g,chi] = znchar(Mod(137, 2021))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2021.137");
 

Basic properties

Modulus: \(2021\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2021\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(322\)
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 2021.ba

\(\chi_{2021}(22,\cdot)\) \(\chi_{2021}(39,\cdot)\) \(\chi_{2021}(45,\cdot)\) \(\chi_{2021}(70,\cdot)\) \(\chi_{2021}(82,\cdot)\) \(\chi_{2021}(88,\cdot)\) \(\chi_{2021}(113,\cdot)\) \(\chi_{2021}(125,\cdot)\) \(\chi_{2021}(137,\cdot)\) \(\chi_{2021}(151,\cdot)\) \(\chi_{2021}(156,\cdot)\) \(\chi_{2021}(161,\cdot)\) \(\chi_{2021}(174,\cdot)\) \(\chi_{2021}(180,\cdot)\) \(\chi_{2021}(199,\cdot)\) \(\chi_{2021}(211,\cdot)\) \(\chi_{2021}(217,\cdot)\) \(\chi_{2021}(223,\cdot)\) \(\chi_{2021}(254,\cdot)\) \(\chi_{2021}(266,\cdot)\) \(\chi_{2021}(280,\cdot)\) \(\chi_{2021}(297,\cdot)\) \(\chi_{2021}(323,\cdot)\) \(\chi_{2021}(340,\cdot)\) \(\chi_{2021}(352,\cdot)\) \(\chi_{2021}(389,\cdot)\) \(\chi_{2021}(395,\cdot)\) \(\chi_{2021}(409,\cdot)\) \(\chi_{2021}(414,\cdot)\) \(\chi_{2021}(419,\cdot)\) ...

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: $\Q(\zeta_{161})$
Fixed field: Number field defined by a degree 322 polynomial (not computed)

Values on generators

\((1035,1979)\) → \((e\left(\frac{13}{14}\right),e\left(\frac{13}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2021 }(137, a) \) \(1\)\(1\)\(e\left(\frac{51}{322}\right)\)\(e\left(\frac{187}{322}\right)\)\(e\left(\frac{51}{161}\right)\)\(e\left(\frac{80}{161}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{153}{322}\right)\)\(e\left(\frac{26}{161}\right)\)\(e\left(\frac{211}{322}\right)\)\(e\left(\frac{269}{322}\right)\)
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)
 
\( \chi_{ 2021 }(137,a) \;\) at \(\;a = \) e.g. 2