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Properties

Label 3751.bg
Modulus $3751$
Conductor $121$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([8,0])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(342,3751)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3751\)
Conductor: \(121\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(11\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 121.e
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 11.11.672749994932560009201.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{3751}(342,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{3751}(683,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{3751}(1024,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{3751}(1365,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{3751}(1706,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{3751}(2047,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{3751}(2388,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{3751}(2729,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{11}\right)\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{3751}(3070,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{3751}(3411,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\)