Properties

Label 4031.840
Modulus $4031$
Conductor $4031$
Order $46$
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(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,14]))
 
pari: [g,chi] = znchar(Mod(840,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
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.z

\(\chi_{4031}(57,\cdot)\) \(\chi_{4031}(173,\cdot)\) \(\chi_{4031}(202,\cdot)\) \(\chi_{4031}(608,\cdot)\) \(\chi_{4031}(811,\cdot)\) \(\chi_{4031}(840,\cdot)\) \(\chi_{4031}(898,\cdot)\) \(\chi_{4031}(1565,\cdot)\) \(\chi_{4031}(1594,\cdot)\) \(\chi_{4031}(1768,\cdot)\) \(\chi_{4031}(1797,\cdot)\) \(\chi_{4031}(1884,\cdot)\) \(\chi_{4031}(1913,\cdot)\) \(\chi_{4031}(2058,\cdot)\) \(\chi_{4031}(2696,\cdot)\) \(\chi_{4031}(3044,\cdot)\) \(\chi_{4031}(3102,\cdot)\) \(\chi_{4031}(3189,\cdot)\) \(\chi_{4031}(3276,\cdot)\) \(\chi_{4031}(3566,\cdot)\) \(\chi_{4031}(3798,\cdot)\) \(\chi_{4031}(3972,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 46 polynomial

Values on generators

\((3337,697)\) → \((-1,e\left(\frac{7}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(840, a) \) \(1\)\(1\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{19}{46}\right)\)\(e\left(\frac{22}{23}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{29}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(840,a) \;\) at \(\;a = \) e.g. 2