Properties

Label 4029.32
Modulus $4029$
Conductor $4029$
Order $312$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(312))
 
M = H._module
 
chi = DirichletCharacter(H, M([156,117,80]))
 
pari: [g,chi] = znchar(Mod(32,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(312\)
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 4029.cu

\(\chi_{4029}(2,\cdot)\) \(\chi_{4029}(26,\cdot)\) \(\chi_{4029}(32,\cdot)\) \(\chi_{4029}(83,\cdot)\) \(\chi_{4029}(104,\cdot)\) \(\chi_{4029}(110,\cdot)\) \(\chi_{4029}(128,\cdot)\) \(\chi_{4029}(155,\cdot)\) \(\chi_{4029}(230,\cdot)\) \(\chi_{4029}(257,\cdot)\) \(\chi_{4029}(263,\cdot)\) \(\chi_{4029}(281,\cdot)\) \(\chi_{4029}(287,\cdot)\) \(\chi_{4029}(332,\cdot)\) \(\chi_{4029}(365,\cdot)\) \(\chi_{4029}(389,\cdot)\) \(\chi_{4029}(440,\cdot)\) \(\chi_{4029}(467,\cdot)\) \(\chi_{4029}(485,\cdot)\) \(\chi_{4029}(518,\cdot)\) \(\chi_{4029}(569,\cdot)\) \(\chi_{4029}(593,\cdot)\) \(\chi_{4029}(716,\cdot)\) \(\chi_{4029}(722,\cdot)\) \(\chi_{4029}(920,\cdot)\) \(\chi_{4029}(950,\cdot)\) \(\chi_{4029}(1046,\cdot)\) \(\chi_{4029}(1052,\cdot)\) \(\chi_{4029}(1103,\cdot)\) \(\chi_{4029}(1148,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{10}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{121}{156}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{85}{312}\right)\)\(e\left(\frac{223}{312}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{5}{104}\right)\)\(e\left(\frac{175}{312}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{51}{104}\right)\)\(e\left(\frac{4}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(32,a) \;\) at \(\;a = \) e.g. 2