↑

Properties

Label 8732.6489
Modulus $8732$
Conductor $2183$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8732, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,11,6]))
 
Copy content pari:[g,chi] = znchar(Mod(6489,8732))
 

Basic properties

Modulus: \(8732\)
Conductor: \(2183\)
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 \(\chi_{2183}(2123,\cdot)\)
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 8732.y

\(\chi_{8732}(3893,\cdot)\) \(\chi_{8732}(6017,\cdot)\) \(\chi_{8732}(6489,\cdot)\) \(\chi_{8732}(8613,\cdot)\)

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: Number field defined by a degree 12 polynomial

Values on generators

\((4367,1889,297)\) → \((1,e\left(\frac{11}{12}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8732 }(6489, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8732 }(6489,a) \;\) at \(\;a = \) e.g. 2