Properties

Label 1024.47
Modulus $1024$
Conductor $256$
Order $64$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,43]))
 
pari: [g,chi] = znchar(Mod(47,1024))
 

Basic properties

Modulus: \(1024\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1024.n

\(\chi_{1024}(15,\cdot)\) \(\chi_{1024}(47,\cdot)\) \(\chi_{1024}(79,\cdot)\) \(\chi_{1024}(111,\cdot)\) \(\chi_{1024}(143,\cdot)\) \(\chi_{1024}(175,\cdot)\) \(\chi_{1024}(207,\cdot)\) \(\chi_{1024}(239,\cdot)\) \(\chi_{1024}(271,\cdot)\) \(\chi_{1024}(303,\cdot)\) \(\chi_{1024}(335,\cdot)\) \(\chi_{1024}(367,\cdot)\) \(\chi_{1024}(399,\cdot)\) \(\chi_{1024}(431,\cdot)\) \(\chi_{1024}(463,\cdot)\) \(\chi_{1024}(495,\cdot)\) \(\chi_{1024}(527,\cdot)\) \(\chi_{1024}(559,\cdot)\) \(\chi_{1024}(591,\cdot)\) \(\chi_{1024}(623,\cdot)\) \(\chi_{1024}(655,\cdot)\) \(\chi_{1024}(687,\cdot)\) \(\chi_{1024}(719,\cdot)\) \(\chi_{1024}(751,\cdot)\) \(\chi_{1024}(783,\cdot)\) \(\chi_{1024}(815,\cdot)\) \(\chi_{1024}(847,\cdot)\) \(\chi_{1024}(879,\cdot)\) \(\chi_{1024}(911,\cdot)\) \(\chi_{1024}(943,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1023,5)\) → \((-1,e\left(\frac{43}{64}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1024 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{43}{64}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{39}{64}\right)\)\(e\left(\frac{37}{64}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{15}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1024 }(47,a) \;\) at \(\;a = \) e.g. 2