Properties

Label 1029.2
Modulus $1029$
Conductor $1029$
Order $294$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1029, base_ring=CyclotomicField(294))
 
M = H._module
 
chi = DirichletCharacter(H, M([147,194]))
 
pari: [g,chi] = znchar(Mod(2,1029))
 

Basic properties

Modulus: \(1029\)
Conductor: \(1029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(294\)
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 1029.v

\(\chi_{1029}(2,\cdot)\) \(\chi_{1029}(11,\cdot)\) \(\chi_{1029}(23,\cdot)\) \(\chi_{1029}(32,\cdot)\) \(\chi_{1029}(44,\cdot)\) \(\chi_{1029}(53,\cdot)\) \(\chi_{1029}(65,\cdot)\) \(\chi_{1029}(74,\cdot)\) \(\chi_{1029}(86,\cdot)\) \(\chi_{1029}(95,\cdot)\) \(\chi_{1029}(107,\cdot)\) \(\chi_{1029}(137,\cdot)\) \(\chi_{1029}(149,\cdot)\) \(\chi_{1029}(158,\cdot)\) \(\chi_{1029}(170,\cdot)\) \(\chi_{1029}(179,\cdot)\) \(\chi_{1029}(191,\cdot)\) \(\chi_{1029}(200,\cdot)\) \(\chi_{1029}(212,\cdot)\) \(\chi_{1029}(221,\cdot)\) \(\chi_{1029}(233,\cdot)\) \(\chi_{1029}(242,\cdot)\) \(\chi_{1029}(254,\cdot)\) \(\chi_{1029}(284,\cdot)\) \(\chi_{1029}(296,\cdot)\) \(\chi_{1029}(305,\cdot)\) \(\chi_{1029}(317,\cdot)\) \(\chi_{1029}(326,\cdot)\) \(\chi_{1029}(338,\cdot)\) \(\chi_{1029}(347,\cdot)\) ...

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: $\Q(\zeta_{147})$
Fixed field: Number field defined by a degree 294 polynomial (not computed)

Values on generators

\((344,346)\) → \((-1,e\left(\frac{97}{147}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1029 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{151}{294}\right)\)\(e\left(\frac{4}{147}\right)\)\(e\left(\frac{187}{294}\right)\)\(e\left(\frac{53}{98}\right)\)\(e\left(\frac{22}{147}\right)\)\(e\left(\frac{221}{294}\right)\)\(e\left(\frac{3}{49}\right)\)\(e\left(\frac{8}{147}\right)\)\(e\left(\frac{293}{294}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1029 }(2,a) \;\) at \(\;a = \) e.g. 2