Properties

Label 1049.1046
Modulus $1049$
Conductor $1049$
Order $1048$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1049, base_ring=CyclotomicField(1048))
 
M = H._module
 
chi = DirichletCharacter(H, M([525]))
 
pari: [g,chi] = znchar(Mod(1046,1049))
 

Basic properties

Modulus: \(1049\)
Conductor: \(1049\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1048\)
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 1049.h

\(\chi_{1049}(3,\cdot)\) \(\chi_{1049}(6,\cdot)\) \(\chi_{1049}(7,\cdot)\) \(\chi_{1049}(12,\cdot)\) \(\chi_{1049}(14,\cdot)\) \(\chi_{1049}(15,\cdot)\) \(\chi_{1049}(17,\cdot)\) \(\chi_{1049}(23,\cdot)\) \(\chi_{1049}(24,\cdot)\) \(\chi_{1049}(27,\cdot)\) \(\chi_{1049}(28,\cdot)\) \(\chi_{1049}(30,\cdot)\) \(\chi_{1049}(31,\cdot)\) \(\chi_{1049}(33,\cdot)\) \(\chi_{1049}(34,\cdot)\) \(\chi_{1049}(35,\cdot)\) \(\chi_{1049}(37,\cdot)\) \(\chi_{1049}(39,\cdot)\) \(\chi_{1049}(41,\cdot)\) \(\chi_{1049}(46,\cdot)\) \(\chi_{1049}(47,\cdot)\) \(\chi_{1049}(48,\cdot)\) \(\chi_{1049}(54,\cdot)\) \(\chi_{1049}(56,\cdot)\) \(\chi_{1049}(57,\cdot)\) \(\chi_{1049}(60,\cdot)\) \(\chi_{1049}(62,\cdot)\) \(\chi_{1049}(63,\cdot)\) \(\chi_{1049}(66,\cdot)\) \(\chi_{1049}(67,\cdot)\) ...

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: $\Q(\zeta_{1048})$
Fixed field: Number field defined by a degree 1048 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{525}{1048}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1049 }(1046, a) \) \(-1\)\(1\)\(e\left(\frac{149}{262}\right)\)\(e\left(\frac{525}{1048}\right)\)\(e\left(\frac{18}{131}\right)\)\(e\left(\frac{49}{524}\right)\)\(e\left(\frac{73}{1048}\right)\)\(e\left(\frac{779}{1048}\right)\)\(e\left(\frac{185}{262}\right)\)\(e\left(\frac{1}{524}\right)\)\(e\left(\frac{347}{524}\right)\)\(e\left(\frac{209}{524}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1049 }(1046,a) \;\) at \(\;a = \) e.g. 2