Properties

Label 1033.27
Modulus $1033$
Conductor $1033$
Order $172$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1033, base_ring=CyclotomicField(172))
 
M = H._module
 
chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(27,1033))
 

Basic properties

Modulus: \(1033\)
Conductor: \(1033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(172\)
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 1033.l

\(\chi_{1033}(6,\cdot)\) \(\chi_{1033}(7,\cdot)\) \(\chi_{1033}(27,\cdot)\) \(\chi_{1033}(34,\cdot)\) \(\chi_{1033}(48,\cdot)\) \(\chi_{1033}(50,\cdot)\) \(\chi_{1033}(56,\cdot)\) \(\chi_{1033}(61,\cdot)\) \(\chi_{1033}(62,\cdot)\) \(\chi_{1033}(101,\cdot)\) \(\chi_{1033}(110,\cdot)\) \(\chi_{1033}(130,\cdot)\) \(\chi_{1033}(153,\cdot)\) \(\chi_{1033}(164,\cdot)\) \(\chi_{1033}(166,\cdot)\) \(\chi_{1033}(172,\cdot)\) \(\chi_{1033}(191,\cdot)\) \(\chi_{1033}(216,\cdot)\) \(\chi_{1033}(222,\cdot)\) \(\chi_{1033}(225,\cdot)\) \(\chi_{1033}(228,\cdot)\) \(\chi_{1033}(242,\cdot)\) \(\chi_{1033}(252,\cdot)\) \(\chi_{1033}(254,\cdot)\) \(\chi_{1033}(259,\cdot)\) \(\chi_{1033}(266,\cdot)\) \(\chi_{1033}(272,\cdot)\) \(\chi_{1033}(279,\cdot)\) \(\chi_{1033}(286,\cdot)\) \(\chi_{1033}(290,\cdot)\) ...

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: $\Q(\zeta_{172})$
Fixed field: Number field defined by a degree 172 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{7}{172}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1033 }(27, a) \) \(1\)\(1\)\(e\left(\frac{35}{43}\right)\)\(e\left(\frac{49}{86}\right)\)\(e\left(\frac{27}{43}\right)\)\(e\left(\frac{7}{172}\right)\)\(e\left(\frac{33}{86}\right)\)\(e\left(\frac{59}{86}\right)\)\(e\left(\frac{19}{43}\right)\)\(e\left(\frac{6}{43}\right)\)\(e\left(\frac{147}{172}\right)\)\(e\left(\frac{171}{172}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1033 }(27,a) \;\) at \(\;a = \) e.g. 2