Properties

Label 1024.21
Modulus $1024$
Conductor $1024$
Order $256$
Real no
Primitive yes
Minimal yes
Parity even

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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([0,109]))
 
pari: [g,chi] = znchar(Mod(21,1024))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1024.q

\(\chi_{1024}(5,\cdot)\) \(\chi_{1024}(13,\cdot)\) \(\chi_{1024}(21,\cdot)\) \(\chi_{1024}(29,\cdot)\) \(\chi_{1024}(37,\cdot)\) \(\chi_{1024}(45,\cdot)\) \(\chi_{1024}(53,\cdot)\) \(\chi_{1024}(61,\cdot)\) \(\chi_{1024}(69,\cdot)\) \(\chi_{1024}(77,\cdot)\) \(\chi_{1024}(85,\cdot)\) \(\chi_{1024}(93,\cdot)\) \(\chi_{1024}(101,\cdot)\) \(\chi_{1024}(109,\cdot)\) \(\chi_{1024}(117,\cdot)\) \(\chi_{1024}(125,\cdot)\) \(\chi_{1024}(133,\cdot)\) \(\chi_{1024}(141,\cdot)\) \(\chi_{1024}(149,\cdot)\) \(\chi_{1024}(157,\cdot)\) \(\chi_{1024}(165,\cdot)\) \(\chi_{1024}(173,\cdot)\) \(\chi_{1024}(181,\cdot)\) \(\chi_{1024}(189,\cdot)\) \(\chi_{1024}(197,\cdot)\) \(\chi_{1024}(205,\cdot)\) \(\chi_{1024}(213,\cdot)\) \(\chi_{1024}(221,\cdot)\) \(\chi_{1024}(229,\cdot)\) \(\chi_{1024}(237,\cdot)\) ...

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

\((1023,5)\) → \((1,e\left(\frac{109}{256}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1024 }(21, a) \) \(1\)\(1\)\(e\left(\frac{103}{256}\right)\)\(e\left(\frac{109}{256}\right)\)\(e\left(\frac{1}{128}\right)\)\(e\left(\frac{103}{128}\right)\)\(e\left(\frac{177}{256}\right)\)\(e\left(\frac{195}{256}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{27}{64}\right)\)\(e\left(\frac{75}{256}\right)\)\(e\left(\frac{105}{256}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1024 }(21,a) \;\) at \(\;a = \) e.g. 2