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Properties

Label 7163.4442
Modulus $7163$
Conductor $7163$
Order $126$
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(7163, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([84,77,99]))
 
Copy content gp:[g,chi] = znchar(Mod(4442, 7163))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7163.4442");
 

Basic properties

Modulus: \(7163\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7163\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(126\)
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 7163.kf

\(\chi_{7163}(295,\cdot)\) \(\chi_{7163}(412,\cdot)\) \(\chi_{7163}(497,\cdot)\) \(\chi_{7163}(705,\cdot)\) \(\chi_{7163}(789,\cdot)\) \(\chi_{7163}(991,\cdot)\) \(\chi_{7163}(1153,\cdot)\) \(\chi_{7163}(1173,\cdot)\) \(\chi_{7163}(1231,\cdot)\) \(\chi_{7163}(1485,\cdot)\) \(\chi_{7163}(1530,\cdot)\) \(\chi_{7163}(1894,\cdot)\) \(\chi_{7163}(2226,\cdot)\) \(\chi_{7163}(2271,\cdot)\) \(\chi_{7163}(2681,\cdot)\) \(\chi_{7163}(2967,\cdot)\) \(\chi_{7163}(3870,\cdot)\) \(\chi_{7163}(3890,\cdot)\) \(\chi_{7163}(3948,\cdot)\) \(\chi_{7163}(4247,\cdot)\) \(\chi_{7163}(4384,\cdot)\) \(\chi_{7163}(4442,\cdot)\) \(\chi_{7163}(4878,\cdot)\) \(\chi_{7163}(4936,\cdot)\) \(\chi_{7163}(4943,\cdot)\) \(\chi_{7163}(5398,\cdot)\) \(\chi_{7163}(5619,\cdot)\) \(\chi_{7163}(5677,\cdot)\) \(\chi_{7163}(5892,\cdot)\) \(\chi_{7163}(6360,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((4409,2263,495)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{18}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 7163 }(4442, a) \) \(-1\)\(1\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{34}{63}\right)\)\(e\left(\frac{8}{63}\right)\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{38}{63}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{8}{63}\right)\)\(e\left(\frac{9}{14}\right)\)
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)
 
\( \chi_{ 7163 }(4442,a) \;\) at \(\;a = \) e.g. 2