Properties

Label 8007.26
Modulus $8007$
Conductor $8007$
Order $312$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, base_ring=CyclotomicField(312))
 
M = H._module
 
chi = DirichletCharacter(H, M([156,39,22]))
 
pari: [g,chi] = znchar(Mod(26,8007))
 

Basic properties

Modulus: \(8007\)
Conductor: \(8007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(312\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8007.fe

\(\chi_{8007}(26,\cdot)\) \(\chi_{8007}(77,\cdot)\) \(\chi_{8007}(104,\cdot)\) \(\chi_{8007}(308,\cdot)\) \(\chi_{8007}(338,\cdot)\) \(\chi_{8007}(416,\cdot)\) \(\chi_{8007}(791,\cdot)\) \(\chi_{8007}(995,\cdot)\) \(\chi_{8007}(1022,\cdot)\) \(\chi_{8007}(1073,\cdot)\) \(\chi_{8007}(1154,\cdot)\) \(\chi_{8007}(1232,\cdot)\) \(\chi_{8007}(1277,\cdot)\) \(\chi_{8007}(1352,\cdot)\) \(\chi_{8007}(1613,\cdot)\) \(\chi_{8007}(1658,\cdot)\) \(\chi_{8007}(1664,\cdot)\) \(\chi_{8007}(1709,\cdot)\) \(\chi_{8007}(1742,\cdot)\) \(\chi_{8007}(1787,\cdot)\) \(\chi_{8007}(1793,\cdot)\) \(\chi_{8007}(1811,\cdot)\) \(\chi_{8007}(1889,\cdot)\) \(\chi_{8007}(1946,\cdot)\) \(\chi_{8007}(2021,\cdot)\) \(\chi_{8007}(2270,\cdot)\) \(\chi_{8007}(2321,\cdot)\) \(\chi_{8007}(2429,\cdot)\) \(\chi_{8007}(2450,\cdot)\) \(\chi_{8007}(2474,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((5339,1414,7855)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{11}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8007 }(26, a) \) \(1\)\(1\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{61}{312}\right)\)\(e\left(\frac{77}{104}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{121}{312}\right)\)\(e\left(\frac{109}{312}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{97}{104}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8007 }(26,a) \;\) at \(\;a = \) e.g. 2