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Properties

Label 6400.4321
Modulus $6400$
Conductor $800$
Order $40$
Real no
Primitive no
Minimal no
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(6400, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([0,5,24]))
 
Copy content gp:[g,chi] = znchar(Mod(4321, 6400))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6400.4321");
 

Basic properties

Modulus: \(6400\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(800\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(40\)
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: no, induced from \(\chi_{800}(421,\cdot)\)
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: no
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 6400.cq

\(\chi_{6400}(161,\cdot)\) \(\chi_{6400}(481,\cdot)\) \(\chi_{6400}(1121,\cdot)\) \(\chi_{6400}(1441,\cdot)\) \(\chi_{6400}(1761,\cdot)\) \(\chi_{6400}(2081,\cdot)\) \(\chi_{6400}(2721,\cdot)\) \(\chi_{6400}(3041,\cdot)\) \(\chi_{6400}(3361,\cdot)\) \(\chi_{6400}(3681,\cdot)\) \(\chi_{6400}(4321,\cdot)\) \(\chi_{6400}(4641,\cdot)\) \(\chi_{6400}(4961,\cdot)\) \(\chi_{6400}(5281,\cdot)\) \(\chi_{6400}(5921,\cdot)\) \(\chi_{6400}(6241,\cdot)\)

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: \(\Q(\zeta_{40})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 40 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((4351,4101,5377)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 6400 }(4321, a) \) \(1\)\(1\)\(e\left(\frac{23}{40}\right)\)\(i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{29}{40}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 6400 }(4321,a) \;\) at \(\;a = \) e.g. 2