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Properties

Label 4056.cg
Modulus $4056$
Conductor $1352$
Order $26$
Real no
Primitive no
Minimal yes
Parity even

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(4056, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([0,13,0,8])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(157, 4056)) 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("4056.157"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(4056\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1352\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(26\)
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 1352.bf
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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 26 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{4056}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\)
\(\chi_{4056}(469,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\)
\(\chi_{4056}(781,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\)
\(\chi_{4056}(1093,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\)
\(\chi_{4056}(1405,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\)
\(\chi_{4056}(1717,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\)
\(\chi_{4056}(2341,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\)
\(\chi_{4056}(2653,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\)
\(\chi_{4056}(2965,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\)
\(\chi_{4056}(3277,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\)
\(\chi_{4056}(3589,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\)
\(\chi_{4056}(3901,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\)