Properties

Label 4031.416
Modulus $4031$
Conductor $4031$
Order $28$
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(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,14]))
 
pari: [g,chi] = znchar(Mod(416,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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.s

\(\chi_{4031}(416,\cdot)\) \(\chi_{4031}(694,\cdot)\) \(\chi_{4031}(833,\cdot)\) \(\chi_{4031}(972,\cdot)\) \(\chi_{4031}(1250,\cdot)\) \(\chi_{4031}(1389,\cdot)\) \(\chi_{4031}(1667,\cdot)\) \(\chi_{4031}(1806,\cdot)\) \(\chi_{4031}(1945,\cdot)\) \(\chi_{4031}(2223,\cdot)\) \(\chi_{4031}(2918,\cdot)\) \(\chi_{4031}(3752,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((3337,697)\) → \((e\left(\frac{23}{28}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(416, a) \) \(1\)\(1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{15}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(416,a) \;\) at \(\;a = \) e.g. 2