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Properties

Label 5328.769
Modulus $5328$
Conductor $333$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,0,4,7]))
 
Copy content pari:[g,chi] = znchar(Mod(769,5328))
 

Basic properties

Modulus: \(5328\)
Conductor: \(333\)
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 \(\chi_{333}(103,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 5328.he

\(\chi_{5328}(97,\cdot)\) \(\chi_{5328}(193,\cdot)\) \(\chi_{5328}(769,\cdot)\) \(\chi_{5328}(4417,\cdot)\)

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: 12.0.7658770225723955928955773.1

Values on generators

\((1999,1333,2369,1297)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5328 }(769, a) \) \(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5328 }(769,a) \;\) at \(\;a = \) e.g. 2