Properties

Label 167.145
Modulus $167$
Conductor $167$
Order $166$
Real no
Primitive yes
Minimal yes
Parity odd

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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([151]))
 
pari: [g,chi] = znchar(Mod(145,167))
 

Basic properties

Modulus: \(167\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 167.d

\(\chi_{167}(5,\cdot)\) \(\chi_{167}(10,\cdot)\) \(\chi_{167}(13,\cdot)\) \(\chi_{167}(15,\cdot)\) \(\chi_{167}(17,\cdot)\) \(\chi_{167}(20,\cdot)\) \(\chi_{167}(23,\cdot)\) \(\chi_{167}(26,\cdot)\) \(\chi_{167}(30,\cdot)\) \(\chi_{167}(34,\cdot)\) \(\chi_{167}(35,\cdot)\) \(\chi_{167}(37,\cdot)\) \(\chi_{167}(39,\cdot)\) \(\chi_{167}(40,\cdot)\) \(\chi_{167}(41,\cdot)\) \(\chi_{167}(43,\cdot)\) \(\chi_{167}(45,\cdot)\) \(\chi_{167}(46,\cdot)\) \(\chi_{167}(51,\cdot)\) \(\chi_{167}(52,\cdot)\) \(\chi_{167}(53,\cdot)\) \(\chi_{167}(55,\cdot)\) \(\chi_{167}(59,\cdot)\) \(\chi_{167}(60,\cdot)\) \(\chi_{167}(67,\cdot)\) \(\chi_{167}(68,\cdot)\) \(\chi_{167}(69,\cdot)\) \(\chi_{167}(70,\cdot)\) \(\chi_{167}(71,\cdot)\) \(\chi_{167}(73,\cdot)\) ...

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 166 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{151}{166}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 167 }(145, a) \) \(-1\)\(1\)\(e\left(\frac{32}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{64}{83}\right)\)\(e\left(\frac{151}{166}\right)\)\(e\left(\frac{74}{83}\right)\)\(e\left(\frac{28}{83}\right)\)\(e\left(\frac{13}{83}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{49}{166}\right)\)\(e\left(\frac{39}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 167 }(145,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 167 }(145,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 167 }(145,·),\chi_{ 167 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 167 }(145,·)) \;\) at \(\; a,b = \) e.g. 1,2