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Properties

Label 3024.163
Modulus $3024$
Conductor $112$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,9,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(163,3024))
 

Basic properties

Modulus: \(3024\)
Conductor: \(112\)
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 \(\chi_{112}(51,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3024.dw

\(\chi_{3024}(163,\cdot)\) \(\chi_{3024}(1243,\cdot)\) \(\chi_{3024}(1675,\cdot)\) \(\chi_{3024}(2755,\cdot)\)

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: 12.0.49519263525896192.1

Values on generators

\((1135,757,785,2593)\) → \((-1,-i,1,e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(163, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(163,a) \;\) at \(\;a = \) e.g. 2