Properties

Label 4031.36
Modulus $4031$
Conductor $4031$
Order $161$
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([138,196]))
 
pari: [g,chi] = znchar(Mod(36,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(161\)
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.bi

\(\chi_{4031}(36,\cdot)\) \(\chi_{4031}(45,\cdot)\) \(\chi_{4031}(52,\cdot)\) \(\chi_{4031}(65,\cdot)\) \(\chi_{4031}(112,\cdot)\) \(\chi_{4031}(194,\cdot)\) \(\chi_{4031}(219,\cdot)\) \(\chi_{4031}(239,\cdot)\) \(\chi_{4031}(255,\cdot)\) \(\chi_{4031}(268,\cdot)\) \(\chi_{4031}(284,\cdot)\) \(\chi_{4031}(314,\cdot)\) \(\chi_{4031}(335,\cdot)\) \(\chi_{4031}(342,\cdot)\) \(\chi_{4031}(343,\cdot)\) \(\chi_{4031}(355,\cdot)\) \(\chi_{4031}(384,\cdot)\) \(\chi_{4031}(451,\cdot)\) \(\chi_{4031}(480,\cdot)\) \(\chi_{4031}(517,\cdot)\) \(\chi_{4031}(529,\cdot)\) \(\chi_{4031}(542,\cdot)\) \(\chi_{4031}(546,\cdot)\) \(\chi_{4031}(600,\cdot)\) \(\chi_{4031}(633,\cdot)\) \(\chi_{4031}(662,\cdot)\) \(\chi_{4031}(687,\cdot)\) \(\chi_{4031}(750,\cdot)\) \(\chi_{4031}(774,\cdot)\) \(\chi_{4031}(807,\cdot)\) ...

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 161 polynomial (not computed)

Values on generators

\((3337,697)\) → \((e\left(\frac{3}{7}\right),e\left(\frac{14}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(36, a) \) \(1\)\(1\)\(e\left(\frac{6}{161}\right)\)\(e\left(\frac{16}{161}\right)\)\(e\left(\frac{12}{161}\right)\)\(e\left(\frac{125}{161}\right)\)\(e\left(\frac{22}{161}\right)\)\(e\left(\frac{93}{161}\right)\)\(e\left(\frac{18}{161}\right)\)\(e\left(\frac{32}{161}\right)\)\(e\left(\frac{131}{161}\right)\)\(e\left(\frac{157}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(36,a) \;\) at \(\;a = \) e.g. 2