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Properties

Label 8112.319
Modulus $8112$
Conductor $52$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,0,0,11]))
 
Copy content pari:[g,chi] = znchar(Mod(319,8112))
 

Basic properties

Modulus: \(8112\)
Conductor: \(52\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{52}(7,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 8112.cq

\(\chi_{8112}(319,\cdot)\) \(\chi_{8112}(2047,\cdot)\) \(\chi_{8112}(3967,\cdot)\) \(\chi_{8112}(6511,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{52})^+\)

Values on generators

\((5071,6085,2705,3889)\) → \((-1,1,1,e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8112 }(319, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8112 }(319,a) \;\) at \(\;a = \) e.g. 2