Properties

Label 8027.53
Modulus $8027$
Conductor $8027$
Order $1276$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(1276))
 
M = H._module
 
chi = DirichletCharacter(H, M([1102,1111]))
 
pari: [g,chi] = znchar(Mod(53,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1276\)
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 8027.bq

\(\chi_{8027}(10,\cdot)\) \(\chi_{8027}(11,\cdot)\) \(\chi_{8027}(21,\cdot)\) \(\chi_{8027}(28,\cdot)\) \(\chi_{8027}(38,\cdot)\) \(\chi_{8027}(53,\cdot)\) \(\chi_{8027}(61,\cdot)\) \(\chi_{8027}(65,\cdot)\) \(\chi_{8027}(79,\cdot)\) \(\chi_{8027}(102,\cdot)\) \(\chi_{8027}(103,\cdot)\) \(\chi_{8027}(182,\cdot)\) \(\chi_{8027}(203,\cdot)\) \(\chi_{8027}(218,\cdot)\) \(\chi_{8027}(222,\cdot)\) \(\chi_{8027}(247,\cdot)\) \(\chi_{8027}(251,\cdot)\) \(\chi_{8027}(270,\cdot)\) \(\chi_{8027}(291,\cdot)\) \(\chi_{8027}(296,\cdot)\) \(\chi_{8027}(297,\cdot)\) \(\chi_{8027}(310,\cdot)\) \(\chi_{8027}(314,\cdot)\) \(\chi_{8027}(339,\cdot)\) \(\chi_{8027}(341,\cdot)\) \(\chi_{8027}(343,\cdot)\) \(\chi_{8027}(355,\cdot)\) \(\chi_{8027}(359,\cdot)\) \(\chi_{8027}(360,\cdot)\) \(\chi_{8027}(387,\cdot)\) ...

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: $\Q(\zeta_{1276})$
Fixed field: Number field defined by a degree 1276 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{19}{22}\right),e\left(\frac{101}{116}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(53, a) \) \(1\)\(1\)\(e\left(\frac{763}{1276}\right)\)\(e\left(\frac{291}{638}\right)\)\(e\left(\frac{125}{638}\right)\)\(e\left(\frac{83}{319}\right)\)\(e\left(\frac{69}{1276}\right)\)\(e\left(\frac{951}{1276}\right)\)\(e\left(\frac{1013}{1276}\right)\)\(e\left(\frac{291}{319}\right)\)\(e\left(\frac{1095}{1276}\right)\)\(e\left(\frac{73}{1276}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(53,a) \;\) at \(\;a = \) e.g. 2