Properties

Label 4031.829
Modulus $4031$
Conductor $4031$
Order $276$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, base_ring=CyclotomicField(276))
 
M = H._module
 
chi = DirichletCharacter(H, M([207,34]))
 
pari: [g,chi] = znchar(Mod(829,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(276\)
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 4031.bk

\(\chi_{4031}(12,\cdot)\) \(\chi_{4031}(17,\cdot)\) \(\chi_{4031}(70,\cdot)\) \(\chi_{4031}(104,\cdot)\) \(\chi_{4031}(128,\cdot)\) \(\chi_{4031}(157,\cdot)\) \(\chi_{4031}(249,\cdot)\) \(\chi_{4031}(273,\cdot)\) \(\chi_{4031}(331,\cdot)\) \(\chi_{4031}(336,\cdot)\) \(\chi_{4031}(389,\cdot)\) \(\chi_{4031}(505,\cdot)\) \(\chi_{4031}(510,\cdot)\) \(\chi_{4031}(568,\cdot)\) \(\chi_{4031}(626,\cdot)\) \(\chi_{4031}(679,\cdot)\) \(\chi_{4031}(684,\cdot)\) \(\chi_{4031}(713,\cdot)\) \(\chi_{4031}(829,\cdot)\) \(\chi_{4031}(853,\cdot)\) \(\chi_{4031}(887,\cdot)\) \(\chi_{4031}(945,\cdot)\) \(\chi_{4031}(969,\cdot)\) \(\chi_{4031}(1061,\cdot)\) \(\chi_{4031}(1114,\cdot)\) \(\chi_{4031}(1235,\cdot)\) \(\chi_{4031}(1409,\cdot)\) \(\chi_{4031}(1462,\cdot)\) \(\chi_{4031}(1491,\cdot)\) \(\chi_{4031}(1520,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((3337,697)\) → \((-i,e\left(\frac{17}{138}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(829, a) \) \(1\)\(1\)\(e\left(\frac{241}{276}\right)\)\(e\left(\frac{221}{276}\right)\)\(e\left(\frac{103}{138}\right)\)\(e\left(\frac{13}{138}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{11}{69}\right)\)\(e\left(\frac{57}{92}\right)\)\(e\left(\frac{83}{138}\right)\)\(e\left(\frac{89}{92}\right)\)\(e\left(\frac{31}{276}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(829,a) \;\) at \(\;a = \) e.g. 2