Properties

Label 4592.407
Modulus $4592$
Conductor $328$
Order $8$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4592, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4,0,7]))
 
pari: [g,chi] = znchar(Mod(407,4592))
 

Basic properties

Modulus: \(4592\)
Conductor: \(328\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{328}(243,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4592.cp

\(\chi_{4592}(407,\cdot)\) \(\chi_{4592}(519,\cdot)\) \(\chi_{4592}(3991,\cdot)\) \(\chi_{4592}(4103,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.797713505816576.1

Values on generators

\((575,3445,3937,785)\) → \((-1,-1,1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 4592 }(407, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4592 }(407,a) \;\) at \(\;a = \) e.g. 2