Properties

Label 4031.332
Modulus $4031$
Conductor $4031$
Order $966$
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(966))
 
M = H._module
 
chi = DirichletCharacter(H, M([621,868]))
 
pari: [g,chi] = znchar(Mod(332,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(966\)
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.bt

\(\chi_{4031}(4,\cdot)\) \(\chi_{4031}(5,\cdot)\) \(\chi_{4031}(9,\cdot)\) \(\chi_{4031}(13,\cdot)\) \(\chi_{4031}(35,\cdot)\) \(\chi_{4031}(38,\cdot)\) \(\chi_{4031}(51,\cdot)\) \(\chi_{4031}(67,\cdot)\) \(\chi_{4031}(71,\cdot)\) \(\chi_{4031}(120,\cdot)\) \(\chi_{4031}(121,\cdot)\) \(\chi_{4031}(122,\cdot)\) \(\chi_{4031}(150,\cdot)\) \(\chi_{4031}(167,\cdot)\) \(\chi_{4031}(180,\cdot)\) \(\chi_{4031}(208,\cdot)\) \(\chi_{4031}(225,\cdot)\) \(\chi_{4031}(238,\cdot)\) \(\chi_{4031}(266,\cdot)\) \(\chi_{4031}(283,\cdot)\) \(\chi_{4031}(294,\cdot)\) \(\chi_{4031}(303,\cdot)\) \(\chi_{4031}(324,\cdot)\) \(\chi_{4031}(325,\cdot)\) \(\chi_{4031}(332,\cdot)\) \(\chi_{4031}(361,\cdot)\) \(\chi_{4031}(399,\cdot)\) \(\chi_{4031}(415,\cdot)\) \(\chi_{4031}(428,\cdot)\) \(\chi_{4031}(441,\cdot)\) ...

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: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\((3337,697)\) → \((e\left(\frac{9}{14}\right),e\left(\frac{62}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(332, a) \) \(1\)\(1\)\(e\left(\frac{523}{966}\right)\)\(e\left(\frac{53}{966}\right)\)\(e\left(\frac{40}{483}\right)\)\(e\left(\frac{202}{483}\right)\)\(e\left(\frac{96}{161}\right)\)\(e\left(\frac{310}{483}\right)\)\(e\left(\frac{201}{322}\right)\)\(e\left(\frac{53}{483}\right)\)\(e\left(\frac{309}{322}\right)\)\(e\left(\frac{349}{966}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(332,a) \;\) at \(\;a = \) e.g. 2