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Properties

Label 10304.dx
Modulus $10304$
Conductor $448$
Order $48$
Real no
Primitive no
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(10304, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,3,8,0])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(507, 10304)) 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("10304.507"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(10304\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(448\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(48\)
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 448.bm
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: \(\Q(\zeta_{48})\)
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 48 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(25\) \(27\)
\(\chi_{10304}(507,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{10304}(691,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{9}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{16}\right)\)
\(\chi_{10304}(1795,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{10304}(1979,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{10304}(3083,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{3}{16}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{10304}(3267,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{10304}(4371,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{10304}(4555,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{3}{16}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{10304}(5659,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{15}{16}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{10304}(5843,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{10304}(6947,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{13}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{3}{16}\right)\)
\(\chi_{10304}(7131,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{15}{16}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{10304}(8235,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{16}\right)\)
\(\chi_{10304}(8419,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{13}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{3}{16}\right)\)
\(\chi_{10304}(9523,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{9}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{7}{16}\right)\)
\(\chi_{10304}(9707,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{5}{16}\right)\)