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Properties

Label 8045.7
Modulus $8045$
Conductor $8045$
Order $1608$
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(8045, base_ring=CyclotomicField(1608)) M = H._module chi = DirichletCharacter(H, M([402,1]))
 
Copy content gp:[g,chi] = znchar(Mod(7, 8045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8045.7");
 

Basic properties

Modulus: \(8045\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8045\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1608\)
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 8045.cc

\(\chi_{8045}(7,\cdot)\) \(\chi_{8045}(17,\cdot)\) \(\chi_{8045}(37,\cdot)\) \(\chi_{8045}(43,\cdot)\) \(\chi_{8045}(53,\cdot)\) \(\chi_{8045}(77,\cdot)\) \(\chi_{8045}(83,\cdot)\) \(\chi_{8045}(163,\cdot)\) \(\chi_{8045}(168,\cdot)\) \(\chi_{8045}(188,\cdot)\) \(\chi_{8045}(202,\cdot)\) \(\chi_{8045}(228,\cdot)\) \(\chi_{8045}(232,\cdot)\) \(\chi_{8045}(233,\cdot)\) \(\chi_{8045}(247,\cdot)\) \(\chi_{8045}(252,\cdot)\) \(\chi_{8045}(253,\cdot)\) \(\chi_{8045}(272,\cdot)\) \(\chi_{8045}(277,\cdot)\) \(\chi_{8045}(282,\cdot)\) \(\chi_{8045}(303,\cdot)\) \(\chi_{8045}(337,\cdot)\) \(\chi_{8045}(342,\cdot)\) \(\chi_{8045}(367,\cdot)\) \(\chi_{8045}(368,\cdot)\) \(\chi_{8045}(377,\cdot)\) \(\chi_{8045}(388,\cdot)\) \(\chi_{8045}(407,\cdot)\) \(\chi_{8045}(408,\cdot)\) \(\chi_{8045}(443,\cdot)\) ...

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: $\Q(\zeta_{1608})$
Fixed field: Number field defined by a degree 1608 polynomial (not computed)

Values on generators

\((6437,1616)\) → \((i,e\left(\frac{1}{1608}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 8045 }(7, a) \) \(1\)\(1\)\(e\left(\frac{709}{804}\right)\)\(e\left(\frac{129}{134}\right)\)\(e\left(\frac{307}{402}\right)\)\(e\left(\frac{679}{804}\right)\)\(e\left(\frac{403}{1608}\right)\)\(e\left(\frac{173}{268}\right)\)\(e\left(\frac{62}{67}\right)\)\(e\left(\frac{137}{201}\right)\)\(e\left(\frac{146}{201}\right)\)\(e\left(\frac{463}{804}\right)\)
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)
 
\( \chi_{ 8045 }(7,a) \;\) at \(\;a = \) e.g. 2