Properties

Label 4031.133
Modulus $4031$
Conductor $4031$
Order $92$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, base_ring=CyclotomicField(92))
 
M = H._module
 
chi = DirichletCharacter(H, M([69,74]))
 
pari: [g,chi] = znchar(Mod(133,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
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.bd

\(\chi_{4031}(75,\cdot)\) \(\chi_{4031}(133,\cdot)\) \(\chi_{4031}(162,\cdot)\) \(\chi_{4031}(215,\cdot)\) \(\chi_{4031}(244,\cdot)\) \(\chi_{4031}(360,\cdot)\) \(\chi_{4031}(365,\cdot)\) \(\chi_{4031}(476,\cdot)\) \(\chi_{4031}(650,\cdot)\) \(\chi_{4031}(771,\cdot)\) \(\chi_{4031}(800,\cdot)\) \(\chi_{4031}(882,\cdot)\) \(\chi_{4031}(916,\cdot)\) \(\chi_{4031}(1032,\cdot)\) \(\chi_{4031}(1172,\cdot)\) \(\chi_{4031}(1206,\cdot)\) \(\chi_{4031}(1259,\cdot)\) \(\chi_{4031}(1346,\cdot)\) \(\chi_{4031}(1404,\cdot)\) \(\chi_{4031}(1438,\cdot)\) \(\chi_{4031}(1728,\cdot)\) \(\chi_{4031}(1752,\cdot)\) \(\chi_{4031}(1815,\cdot)\) \(\chi_{4031}(1902,\cdot)\) \(\chi_{4031}(1960,\cdot)\) \(\chi_{4031}(2308,\cdot)\) \(\chi_{4031}(2390,\cdot)\) \(\chi_{4031}(2535,\cdot)\) \(\chi_{4031}(2564,\cdot)\) \(\chi_{4031}(2651,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((3337,697)\) → \((-i,e\left(\frac{37}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(133, a) \) \(1\)\(1\)\(e\left(\frac{51}{92}\right)\)\(e\left(\frac{67}{92}\right)\)\(e\left(\frac{5}{46}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{13}{46}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{61}{92}\right)\)\(e\left(\frac{21}{46}\right)\)\(e\left(\frac{21}{92}\right)\)\(e\left(\frac{81}{92}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(133,a) \;\) at \(\;a = \) e.g. 2