Properties

Label 86.21
Modulus $86$
Conductor $43$
Order $7$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(86, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(21,86))
 

Basic properties

Modulus: \(86\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 86.e

\(\chi_{86}(11,\cdot)\) \(\chi_{86}(21,\cdot)\) \(\chi_{86}(35,\cdot)\) \(\chi_{86}(41,\cdot)\) \(\chi_{86}(47,\cdot)\) \(\chi_{86}(59,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.6321363049.1

Values on generators

\(3\) → \(e\left(\frac{6}{7}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 86 }(21, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 86 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 86 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 86 }(21,·),\chi_{ 86 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 86 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2