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Properties

Label 7021.396
Modulus $7021$
Conductor $7021$
Order $1392$
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(7021, base_ring=CyclotomicField(1392)) M = H._module chi = DirichletCharacter(H, M([928,435,264]))
 
Copy content gp:[g,chi] = znchar(Mod(396, 7021))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7021.396");
 

Basic properties

Modulus: \(7021\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7021\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1392\)
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 7021.cz

\(\chi_{7021}(11,\cdot)\) \(\chi_{7021}(23,\cdot)\) \(\chi_{7021}(37,\cdot)\) \(\chi_{7021}(39,\cdot)\) \(\chi_{7021}(44,\cdot)\) \(\chi_{7021}(65,\cdot)\) \(\chi_{7021}(109,\cdot)\) \(\chi_{7021}(114,\cdot)\) \(\chi_{7021}(142,\cdot)\) \(\chi_{7021}(156,\cdot)\) \(\chi_{7021}(158,\cdot)\) \(\chi_{7021}(165,\cdot)\) \(\chi_{7021}(207,\cdot)\) \(\chi_{7021}(214,\cdot)\) \(\chi_{7021}(233,\cdot)\) \(\chi_{7021}(249,\cdot)\) \(\chi_{7021}(275,\cdot)\) \(\chi_{7021}(303,\cdot)\) \(\chi_{7021}(326,\cdot)\) \(\chi_{7021}(333,\cdot)\) \(\chi_{7021}(345,\cdot)\) \(\chi_{7021}(347,\cdot)\) \(\chi_{7021}(368,\cdot)\) \(\chi_{7021}(394,\cdot)\) \(\chi_{7021}(396,\cdot)\) \(\chi_{7021}(401,\cdot)\) \(\chi_{7021}(415,\cdot)\) \(\chi_{7021}(431,\cdot)\) \(\chi_{7021}(436,\cdot)\) \(\chi_{7021}(445,\cdot)\) ...

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: $\Q(\zeta_{1392})$
Fixed field: Number field defined by a degree 1392 polynomial (not computed)

Values on generators

\((1004,5783,120)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{16}\right),e\left(\frac{11}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 7021 }(396, a) \) \(1\)\(1\)\(e\left(\frac{625}{696}\right)\)\(e\left(\frac{643}{1392}\right)\)\(e\left(\frac{277}{348}\right)\)\(e\left(\frac{47}{1392}\right)\)\(e\left(\frac{167}{464}\right)\)\(e\left(\frac{161}{232}\right)\)\(e\left(\frac{643}{696}\right)\)\(e\left(\frac{1297}{1392}\right)\)\(e\left(\frac{829}{1392}\right)\)\(e\left(\frac{359}{1392}\right)\)
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)
 
\( \chi_{ 7021 }(396,a) \;\) at \(\;a = \) e.g. 2