Properties

Label 4031.204
Modulus $4031$
Conductor $139$
Order $23$
Real no
Primitive no
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([0,4]))
 
pari: [g,chi] = znchar(Mod(204,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(139\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{139}(65,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4031.r

\(\chi_{4031}(175,\cdot)\) \(\chi_{4031}(204,\cdot)\) \(\chi_{4031}(378,\cdot)\) \(\chi_{4031}(407,\cdot)\) \(\chi_{4031}(494,\cdot)\) \(\chi_{4031}(523,\cdot)\) \(\chi_{4031}(668,\cdot)\) \(\chi_{4031}(1306,\cdot)\) \(\chi_{4031}(1654,\cdot)\) \(\chi_{4031}(1712,\cdot)\) \(\chi_{4031}(1799,\cdot)\) \(\chi_{4031}(1886,\cdot)\) \(\chi_{4031}(2176,\cdot)\) \(\chi_{4031}(2408,\cdot)\) \(\chi_{4031}(2582,\cdot)\) \(\chi_{4031}(2698,\cdot)\) \(\chi_{4031}(2814,\cdot)\) \(\chi_{4031}(2843,\cdot)\) \(\chi_{4031}(3249,\cdot)\) \(\chi_{4031}(3452,\cdot)\) \(\chi_{4031}(3481,\cdot)\) \(\chi_{4031}(3539,\cdot)\)

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 23 polynomial

Values on generators

\((3337,697)\) → \((1,e\left(\frac{2}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(204, a) \) \(1\)\(1\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{14}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(204,a) \;\) at \(\;a = \) e.g. 2