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Properties

Label 3042.bh
Modulus $3042$
Conductor $169$
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(3042, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([0,21])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(181, 3042)) 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("3042.181"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(3042\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(169\)
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 169.h
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: 26.26.3830224792147131369362629348887201408953937846517364173.1

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{3042}(181,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{3042}(415,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{3042}(649,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{3042}(883,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{3042}(1117,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{13}\right)\)
\(\chi_{3042}(1585,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{3042}(1819,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{3042}(2053,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{3042}(2287,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{3042}(2521,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{3042}(2755,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{10}{13}\right)\)
\(\chi_{3042}(2989,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{13}\right)\)