Properties

Label 167.2
Modulus $167$
Conductor $167$
Order $83$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(167, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([40]))
 
pari: [g,chi] = znchar(Mod(2,167))
 

Basic properties

Modulus: \(167\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
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 167.c

\(\chi_{167}(2,\cdot)\) \(\chi_{167}(3,\cdot)\) \(\chi_{167}(4,\cdot)\) \(\chi_{167}(6,\cdot)\) \(\chi_{167}(7,\cdot)\) \(\chi_{167}(8,\cdot)\) \(\chi_{167}(9,\cdot)\) \(\chi_{167}(11,\cdot)\) \(\chi_{167}(12,\cdot)\) \(\chi_{167}(14,\cdot)\) \(\chi_{167}(16,\cdot)\) \(\chi_{167}(18,\cdot)\) \(\chi_{167}(19,\cdot)\) \(\chi_{167}(21,\cdot)\) \(\chi_{167}(22,\cdot)\) \(\chi_{167}(24,\cdot)\) \(\chi_{167}(25,\cdot)\) \(\chi_{167}(27,\cdot)\) \(\chi_{167}(28,\cdot)\) \(\chi_{167}(29,\cdot)\) \(\chi_{167}(31,\cdot)\) \(\chi_{167}(32,\cdot)\) \(\chi_{167}(33,\cdot)\) \(\chi_{167}(36,\cdot)\) \(\chi_{167}(38,\cdot)\) \(\chi_{167}(42,\cdot)\) \(\chi_{167}(44,\cdot)\) \(\chi_{167}(47,\cdot)\) \(\chi_{167}(48,\cdot)\) \(\chi_{167}(49,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 83 polynomial

Values on generators

\(5\) → \(e\left(\frac{20}{83}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 167 }(2, a) \) \(1\)\(1\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{54}{83}\right)\)\(e\left(\frac{23}{83}\right)\)\(e\left(\frac{20}{83}\right)\)\(e\left(\frac{24}{83}\right)\)\(e\left(\frac{36}{83}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{25}{83}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{62}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 167 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 167 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 167 }(2,·),\chi_{ 167 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 167 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2