Properties

Label 640.br
Modulus $640$
Conductor $640$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(32)) M = H._module chi = DirichletCharacter(H, M([16,29,24])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(43, 640)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("640.43"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(640\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(640\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(32\)
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);
 

Related number fields

Field of values: \(\Q(\zeta_{32})\)
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: 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.1
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{640}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{32}\right)\)
\(\chi_{640}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{32}\right)\)
\(\chi_{640}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{17}{32}\right)\)
\(\chi_{640}(147,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{32}\right)\)
\(\chi_{640}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{21}{32}\right)\)
\(\chi_{640}(227,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{32}\right)\)
\(\chi_{640}(283,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{25}{32}\right)\)
\(\chi_{640}(307,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{32}\right)\)
\(\chi_{640}(363,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{29}{32}\right)\)
\(\chi_{640}(387,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{19}{32}\right)\)
\(\chi_{640}(443,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{32}\right)\)
\(\chi_{640}(467,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{23}{32}\right)\)
\(\chi_{640}(523,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{32}\right)\)
\(\chi_{640}(547,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{27}{32}\right)\)
\(\chi_{640}(603,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{32}\right)\)
\(\chi_{640}(627,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{31}{32}\right)\)