Properties

Label 173.95
Modulus $173$
Conductor $173$
Order $43$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(173, base_ring=CyclotomicField(86))
 
M = H._module
 
chi = DirichletCharacter(H, M([36]))
 
pari: [g,chi] = znchar(Mod(95,173))
 

Basic properties

Modulus: \(173\)
Conductor: \(173\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(43\)
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 173.d

\(\chi_{173}(6,\cdot)\) \(\chi_{173}(10,\cdot)\) \(\chi_{173}(14,\cdot)\) \(\chi_{173}(16,\cdot)\) \(\chi_{173}(22,\cdot)\) \(\chi_{173}(23,\cdot)\) \(\chi_{173}(29,\cdot)\) \(\chi_{173}(36,\cdot)\) \(\chi_{173}(43,\cdot)\) \(\chi_{173}(47,\cdot)\) \(\chi_{173}(51,\cdot)\) \(\chi_{173}(52,\cdot)\) \(\chi_{173}(57,\cdot)\) \(\chi_{173}(60,\cdot)\) \(\chi_{173}(81,\cdot)\) \(\chi_{173}(83,\cdot)\) \(\chi_{173}(84,\cdot)\) \(\chi_{173}(85,\cdot)\) \(\chi_{173}(95,\cdot)\) \(\chi_{173}(96,\cdot)\) \(\chi_{173}(100,\cdot)\) \(\chi_{173}(106,\cdot)\) \(\chi_{173}(109,\cdot)\) \(\chi_{173}(117,\cdot)\) \(\chi_{173}(118,\cdot)\) \(\chi_{173}(119,\cdot)\) \(\chi_{173}(124,\cdot)\) \(\chi_{173}(132,\cdot)\) \(\chi_{173}(133,\cdot)\) \(\chi_{173}(135,\cdot)\) ...

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: $\Q(\zeta_{43})$
Fixed field: Number field defined by a degree 43 polynomial

Values on generators

\(2\) → \(e\left(\frac{18}{43}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 173 }(95, a) \) \(1\)\(1\)\(e\left(\frac{18}{43}\right)\)\(e\left(\frac{13}{43}\right)\)\(e\left(\frac{36}{43}\right)\)\(e\left(\frac{14}{43}\right)\)\(e\left(\frac{31}{43}\right)\)\(e\left(\frac{33}{43}\right)\)\(e\left(\frac{11}{43}\right)\)\(e\left(\frac{26}{43}\right)\)\(e\left(\frac{32}{43}\right)\)\(e\left(\frac{27}{43}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 173 }(95,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 173 }(95,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 173 }(95,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 173 }(95,·)) \;\) at \(\; a,b = \) e.g. 1,2