Properties

Label 4031.1014
Modulus $4031$
Conductor $4031$
Order $138$
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(138))
 
M = H._module
 
chi = DirichletCharacter(H, M([69,32]))
 
pari: [g,chi] = znchar(Mod(1014,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(138\)
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.bf

\(\chi_{4031}(28,\cdot)\) \(\chi_{4031}(86,\cdot)\) \(\chi_{4031}(144,\cdot)\) \(\chi_{4031}(260,\cdot)\) \(\chi_{4031}(289,\cdot)\) \(\chi_{4031}(347,\cdot)\) \(\chi_{4031}(405,\cdot)\) \(\chi_{4031}(463,\cdot)\) \(\chi_{4031}(637,\cdot)\) \(\chi_{4031}(724,\cdot)\) \(\chi_{4031}(869,\cdot)\) \(\chi_{4031}(956,\cdot)\) \(\chi_{4031}(1014,\cdot)\) \(\chi_{4031}(1072,\cdot)\) \(\chi_{4031}(1159,\cdot)\) \(\chi_{4031}(1275,\cdot)\) \(\chi_{4031}(1420,\cdot)\) \(\chi_{4031}(1507,\cdot)\) \(\chi_{4031}(1536,\cdot)\) \(\chi_{4031}(1681,\cdot)\) \(\chi_{4031}(1739,\cdot)\) \(\chi_{4031}(1971,\cdot)\) \(\chi_{4031}(2000,\cdot)\) \(\chi_{4031}(2029,\cdot)\) \(\chi_{4031}(2116,\cdot)\) \(\chi_{4031}(2174,\cdot)\) \(\chi_{4031}(2203,\cdot)\) \(\chi_{4031}(2261,\cdot)\) \(\chi_{4031}(2290,\cdot)\) \(\chi_{4031}(2348,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 138 polynomial (not computed)

Values on generators

\((3337,697)\) → \((-1,e\left(\frac{16}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(1014, a) \) \(1\)\(1\)\(e\left(\frac{101}{138}\right)\)\(e\left(\frac{1}{138}\right)\)\(e\left(\frac{32}{69}\right)\)\(e\left(\frac{65}{69}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{41}{69}\right)\)\(e\left(\frac{9}{46}\right)\)\(e\left(\frac{1}{69}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{17}{138}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(1014,a) \;\) at \(\;a = \) e.g. 2