Properties

Label 4031.513
Modulus $4031$
Conductor $4031$
Order $21$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,14]))
 
pari: [g,chi] = znchar(Mod(513,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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.q

\(\chi_{4031}(181,\cdot)\) \(\chi_{4031}(459,\cdot)\) \(\chi_{4031}(513,\cdot)\) \(\chi_{4031}(1069,\cdot)\) \(\chi_{4031}(1154,\cdot)\) \(\chi_{4031}(1486,\cdot)\) \(\chi_{4031}(1764,\cdot)\) \(\chi_{4031}(1988,\cdot)\) \(\chi_{4031}(2459,\cdot)\) \(\chi_{4031}(3239,\cdot)\) \(\chi_{4031}(3293,\cdot)\) \(\chi_{4031}(3795,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((3337,697)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(513, a) \) \(1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{16}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(513,a) \;\) at \(\;a = \) e.g. 2