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Properties

Label 13680.ws
Modulus $13680$
Conductor $13680$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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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(13680, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([0,27,12,9,32])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(157, 13680)) 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("13680.157"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(13680\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(13680\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
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: odd
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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(23\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{13680}(157,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{13680}(2077,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{23}{36}\right)\) \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{13680}(2797,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{11}{36}\right)\) \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{13680}(3037,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{13680}(3733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{17}{36}\right)\) \(1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{13680}(5413,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{13680}(7837,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{13680}(9517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{19}{36}\right)\) \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{13680}(9733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{13680}(11653,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{13680}(12373,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{13680}(12613,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{18}\right)\)