Properties

Label 4031.216
Modulus $4031$
Conductor $4031$
Order $322$
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(322))
 
M = H._module
 
chi = DirichletCharacter(H, M([207,294]))
 
pari: [g,chi] = znchar(Mod(216,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(322\)
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.bl

\(\chi_{4031}(6,\cdot)\) \(\chi_{4031}(34,\cdot)\) \(\chi_{4031}(63,\cdot)\) \(\chi_{4031}(64,\cdot)\) \(\chi_{4031}(80,\cdot)\) \(\chi_{4031}(91,\cdot)\) \(\chi_{4031}(100,\cdot)\) \(\chi_{4031}(125,\cdot)\) \(\chi_{4031}(129,\cdot)\) \(\chi_{4031}(183,\cdot)\) \(\chi_{4031}(196,\cdot)\) \(\chi_{4031}(216,\cdot)\) \(\chi_{4031}(245,\cdot)\) \(\chi_{4031}(270,\cdot)\) \(\chi_{4031}(312,\cdot)\) \(\chi_{4031}(323,\cdot)\) \(\chi_{4031}(341,\cdot)\) \(\chi_{4031}(357,\cdot)\) \(\chi_{4031}(390,\cdot)\) \(\chi_{4031}(469,\cdot)\) \(\chi_{4031}(497,\cdot)\) \(\chi_{4031}(613,\cdot)\) \(\chi_{4031}(647,\cdot)\) \(\chi_{4031}(672,\cdot)\) \(\chi_{4031}(701,\cdot)\) \(\chi_{4031}(729,\cdot)\) \(\chi_{4031}(731,\cdot)\) \(\chi_{4031}(747,\cdot)\) \(\chi_{4031}(758,\cdot)\) \(\chi_{4031}(759,\cdot)\) ...

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: $\Q(\zeta_{161})$
Fixed field: Number field defined by a degree 322 polynomial (not computed)

Values on generators

\((3337,697)\) → \((e\left(\frac{9}{14}\right),e\left(\frac{21}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(216, a) \) \(1\)\(1\)\(e\left(\frac{179}{322}\right)\)\(e\left(\frac{209}{322}\right)\)\(e\left(\frac{18}{161}\right)\)\(e\left(\frac{107}{161}\right)\)\(e\left(\frac{33}{161}\right)\)\(e\left(\frac{59}{161}\right)\)\(e\left(\frac{215}{322}\right)\)\(e\left(\frac{48}{161}\right)\)\(e\left(\frac{71}{322}\right)\)\(e\left(\frac{149}{322}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(216,a) \;\) at \(\;a = \) e.g. 2