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Properties

Label 8712.5323
Modulus $8712$
Conductor $792$
Order $6$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8712, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([3,3,2,3]))
 
Copy content pari:[g,chi] = znchar(Mod(5323,8712))
 

Basic properties

Modulus: \(8712\)
Conductor: \(792\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(6\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{792}(571,\cdot)\)
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)
 

Galois orbit 8712.z

\(\chi_{8712}(2419,\cdot)\) \(\chi_{8712}(5323,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 6.6.4471137792.1

Values on generators

\((6535,4357,1937,5689)\) → \((-1,-1,e\left(\frac{1}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8712 }(5323, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8712 }(5323,a) \;\) at \(\;a = \) e.g. 2