Properties

Label 1024.3
Modulus $1024$
Conductor $1024$
Order $256$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(256))
 
M = H._module
 
chi = DirichletCharacter(H, M([128,163]))
 
pari: [g,chi] = znchar(Mod(3,1024))
 

Basic properties

Modulus: \(1024\)
Conductor: \(1024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(256\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1024.r

\(\chi_{1024}(3,\cdot)\) \(\chi_{1024}(11,\cdot)\) \(\chi_{1024}(19,\cdot)\) \(\chi_{1024}(27,\cdot)\) \(\chi_{1024}(35,\cdot)\) \(\chi_{1024}(43,\cdot)\) \(\chi_{1024}(51,\cdot)\) \(\chi_{1024}(59,\cdot)\) \(\chi_{1024}(67,\cdot)\) \(\chi_{1024}(75,\cdot)\) \(\chi_{1024}(83,\cdot)\) \(\chi_{1024}(91,\cdot)\) \(\chi_{1024}(99,\cdot)\) \(\chi_{1024}(107,\cdot)\) \(\chi_{1024}(115,\cdot)\) \(\chi_{1024}(123,\cdot)\) \(\chi_{1024}(131,\cdot)\) \(\chi_{1024}(139,\cdot)\) \(\chi_{1024}(147,\cdot)\) \(\chi_{1024}(155,\cdot)\) \(\chi_{1024}(163,\cdot)\) \(\chi_{1024}(171,\cdot)\) \(\chi_{1024}(179,\cdot)\) \(\chi_{1024}(187,\cdot)\) \(\chi_{1024}(195,\cdot)\) \(\chi_{1024}(203,\cdot)\) \(\chi_{1024}(211,\cdot)\) \(\chi_{1024}(219,\cdot)\) \(\chi_{1024}(227,\cdot)\) \(\chi_{1024}(235,\cdot)\) ...

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: $\Q(\zeta_{256})$
Fixed field: Number field defined by a degree 256 polynomial (not computed)

Values on generators

\((1023,5)\) → \((-1,e\left(\frac{163}{256}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1024 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{73}{256}\right)\)\(e\left(\frac{163}{256}\right)\)\(e\left(\frac{15}{128}\right)\)\(e\left(\frac{73}{128}\right)\)\(e\left(\frac{31}{256}\right)\)\(e\left(\frac{45}{256}\right)\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{21}{64}\right)\)\(e\left(\frac{165}{256}\right)\)\(e\left(\frac{103}{256}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1024 }(3,a) \;\) at \(\;a = \) e.g. 2