Properties

Label 8007.392
Modulus $8007$
Conductor $471$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,0,31]))
 
pari: [g,chi] = znchar(Mod(392,8007))
 

Basic properties

Modulus: \(8007\)
Conductor: \(471\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{471}(392,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8007.dk

\(\chi_{8007}(392,\cdot)\) \(\chi_{8007}(494,\cdot)\) \(\chi_{8007}(596,\cdot)\) \(\chi_{8007}(1106,\cdot)\) \(\chi_{8007}(1310,\cdot)\) \(\chi_{8007}(2483,\cdot)\) \(\chi_{8007}(2891,\cdot)\) \(\chi_{8007}(2942,\cdot)\) \(\chi_{8007}(3095,\cdot)\) \(\chi_{8007}(3299,\cdot)\) \(\chi_{8007}(3452,\cdot)\) \(\chi_{8007}(3656,\cdot)\) \(\chi_{8007}(3809,\cdot)\) \(\chi_{8007}(3860,\cdot)\) \(\chi_{8007}(4268,\cdot)\) \(\chi_{8007}(5441,\cdot)\) \(\chi_{8007}(5645,\cdot)\) \(\chi_{8007}(6155,\cdot)\) \(\chi_{8007}(6257,\cdot)\) \(\chi_{8007}(6359,\cdot)\) \(\chi_{8007}(7073,\cdot)\) \(\chi_{8007}(7124,\cdot)\) \(\chi_{8007}(7634,\cdot)\) \(\chi_{8007}(7685,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((5339,1414,7855)\) → \((-1,1,e\left(\frac{31}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8007 }(392, a) \) \(1\)\(1\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{5}{26}\right)\)\(-1\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{3}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8007 }(392,a) \;\) at \(\;a = \) e.g. 2