Properties

Label 4031.876
Modulus $4031$
Conductor $4031$
Order $42$
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([9,28]))
 
pari: [g,chi] = znchar(Mod(876,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.u

\(\chi_{4031}(42,\cdot)\) \(\chi_{4031}(96,\cdot)\) \(\chi_{4031}(876,\cdot)\) \(\chi_{4031}(1347,\cdot)\) \(\chi_{4031}(1571,\cdot)\) \(\chi_{4031}(1849,\cdot)\) \(\chi_{4031}(2181,\cdot)\) \(\chi_{4031}(2266,\cdot)\) \(\chi_{4031}(2822,\cdot)\) \(\chi_{4031}(2876,\cdot)\) \(\chi_{4031}(3154,\cdot)\) \(\chi_{4031}(3571,\cdot)\)

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 42 polynomial

Values on generators

\((3337,697)\) → \((e\left(\frac{3}{14}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(876, a) \) \(1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(876,a) \;\) at \(\;a = \) e.g. 2