Properties

Label 1049.9
Modulus $1049$
Conductor $1049$
Order $524$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1049, base_ring=CyclotomicField(524))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(9,1049))
 

Basic properties

Modulus: \(1049\)
Conductor: \(1049\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(524\)
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 1049.g

\(\chi_{1049}(5,\cdot)\) \(\chi_{1049}(9,\cdot)\) \(\chi_{1049}(10,\cdot)\) \(\chi_{1049}(11,\cdot)\) \(\chi_{1049}(18,\cdot)\) \(\chi_{1049}(20,\cdot)\) \(\chi_{1049}(22,\cdot)\) \(\chi_{1049}(36,\cdot)\) \(\chi_{1049}(40,\cdot)\) \(\chi_{1049}(44,\cdot)\) \(\chi_{1049}(49,\cdot)\) \(\chi_{1049}(51,\cdot)\) \(\chi_{1049}(59,\cdot)\) \(\chi_{1049}(65,\cdot)\) \(\chi_{1049}(72,\cdot)\) \(\chi_{1049}(79,\cdot)\) \(\chi_{1049}(80,\cdot)\) \(\chi_{1049}(88,\cdot)\) \(\chi_{1049}(93,\cdot)\) \(\chi_{1049}(95,\cdot)\) \(\chi_{1049}(98,\cdot)\) \(\chi_{1049}(102,\cdot)\) \(\chi_{1049}(103,\cdot)\) \(\chi_{1049}(105,\cdot)\) \(\chi_{1049}(107,\cdot)\) \(\chi_{1049}(111,\cdot)\) \(\chi_{1049}(113,\cdot)\) \(\chi_{1049}(117,\cdot)\) \(\chi_{1049}(118,\cdot)\) \(\chi_{1049}(123,\cdot)\) ...

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: $\Q(\zeta_{524})$
Fixed field: Number field defined by a degree 524 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1}{524}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1049 }(9, a) \) \(1\)\(1\)\(e\left(\frac{18}{131}\right)\)\(e\left(\frac{1}{524}\right)\)\(e\left(\frac{36}{131}\right)\)\(e\left(\frac{49}{262}\right)\)\(e\left(\frac{73}{524}\right)\)\(e\left(\frac{255}{524}\right)\)\(e\left(\frac{54}{131}\right)\)\(e\left(\frac{1}{262}\right)\)\(e\left(\frac{85}{262}\right)\)\(e\left(\frac{209}{262}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1049 }(9,a) \;\) at \(\;a = \) e.g. 2