Properties

Label 5184.325
Modulus $5184$
Conductor $64$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5184, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,0]))
 
pari: [g,chi] = znchar(Mod(325,5184))
 

Basic properties

Modulus: \(5184\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5184.be

\(\chi_{5184}(325,\cdot)\) \(\chi_{5184}(973,\cdot)\) \(\chi_{5184}(1621,\cdot)\) \(\chi_{5184}(2269,\cdot)\) \(\chi_{5184}(2917,\cdot)\) \(\chi_{5184}(3565,\cdot)\) \(\chi_{5184}(4213,\cdot)\) \(\chi_{5184}(4861,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)

Values on generators

\((2431,325,1217)\) → \((1,e\left(\frac{1}{16}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5184 }(325, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(-i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5184 }(325,a) \;\) at \(\;a = \) e.g. 2