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Properties

Label 3648.dx
Modulus $3648$
Conductor $3648$
Order $48$
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(3648, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,15,24,32])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(11, 3648)) 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("3648.11"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(3648\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3648\)
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: 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: \(\Q(\zeta_{48})\)
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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{3648}(11,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(1\) \(e\left(\frac{5}{48}\right)\)
\(\chi_{3648}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(1\) \(e\left(\frac{7}{48}\right)\)
\(\chi_{3648}(467,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(1\) \(e\left(\frac{23}{48}\right)\)
\(\chi_{3648}(539,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(1\) \(e\left(\frac{25}{48}\right)\)
\(\chi_{3648}(923,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(1\) \(e\left(\frac{41}{48}\right)\)
\(\chi_{3648}(995,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(1\) \(e\left(\frac{43}{48}\right)\)
\(\chi_{3648}(1379,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(1\) \(e\left(\frac{11}{48}\right)\)
\(\chi_{3648}(1451,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(1\) \(e\left(\frac{13}{48}\right)\)
\(\chi_{3648}(1835,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(1\) \(e\left(\frac{29}{48}\right)\)
\(\chi_{3648}(1907,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(1\) \(e\left(\frac{31}{48}\right)\)
\(\chi_{3648}(2291,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(1\) \(e\left(\frac{47}{48}\right)\)
\(\chi_{3648}(2363,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(1\) \(e\left(\frac{1}{48}\right)\)
\(\chi_{3648}(2747,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(1\) \(e\left(\frac{17}{48}\right)\)
\(\chi_{3648}(2819,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(1\) \(e\left(\frac{19}{48}\right)\)
\(\chi_{3648}(3203,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(1\) \(e\left(\frac{35}{48}\right)\)
\(\chi_{3648}(3275,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(1\) \(e\left(\frac{37}{48}\right)\)