Properties

Label 1049.2
Modulus $1049$
Conductor $1049$
Order $262$
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(262))
 
M = H._module
 
chi = DirichletCharacter(H, M([149]))
 
pari: [g,chi] = znchar(Mod(2,1049))
 

Basic properties

Modulus: \(1049\)
Conductor: \(1049\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(262\)
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.f

\(\chi_{1049}(2,\cdot)\) \(\chi_{1049}(8,\cdot)\) \(\chi_{1049}(13,\cdot)\) \(\chi_{1049}(25,\cdot)\) \(\chi_{1049}(29,\cdot)\) \(\chi_{1049}(32,\cdot)\) \(\chi_{1049}(38,\cdot)\) \(\chi_{1049}(42,\cdot)\) \(\chi_{1049}(43,\cdot)\) \(\chi_{1049}(45,\cdot)\) \(\chi_{1049}(52,\cdot)\) \(\chi_{1049}(55,\cdot)\) \(\chi_{1049}(69,\cdot)\) \(\chi_{1049}(73,\cdot)\) \(\chi_{1049}(81,\cdot)\) \(\chi_{1049}(97,\cdot)\) \(\chi_{1049}(99,\cdot)\) \(\chi_{1049}(100,\cdot)\) \(\chi_{1049}(106,\cdot)\) \(\chi_{1049}(116,\cdot)\) \(\chi_{1049}(119,\cdot)\) \(\chi_{1049}(121,\cdot)\) \(\chi_{1049}(122,\cdot)\) \(\chi_{1049}(128,\cdot)\) \(\chi_{1049}(152,\cdot)\) \(\chi_{1049}(163,\cdot)\) \(\chi_{1049}(168,\cdot)\) \(\chi_{1049}(172,\cdot)\) \(\chi_{1049}(180,\cdot)\) \(\chi_{1049}(181,\cdot)\) ...

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: $\Q(\zeta_{131})$
Fixed field: Number field defined by a degree 262 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{149}{262}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1049 }(2, a) \) \(1\)\(1\)\(e\left(\frac{124}{131}\right)\)\(e\left(\frac{149}{262}\right)\)\(e\left(\frac{117}{131}\right)\)\(e\left(\frac{96}{131}\right)\)\(e\left(\frac{135}{262}\right)\)\(e\left(\frac{5}{262}\right)\)\(e\left(\frac{110}{131}\right)\)\(e\left(\frac{18}{131}\right)\)\(e\left(\frac{89}{131}\right)\)\(e\left(\frac{94}{131}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1049 }(2,a) \;\) at \(\;a = \) e.g. 2