Properties

Label 501.17
Modulus $501$
Conductor $501$
Order $166$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(501, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,53]))
 
pari: [g,chi] = znchar(Mod(17,501))
 

Basic properties

Modulus: \(501\)
Conductor: \(501\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 501.g

\(\chi_{501}(5,\cdot)\) \(\chi_{501}(17,\cdot)\) \(\chi_{501}(20,\cdot)\) \(\chi_{501}(23,\cdot)\) \(\chi_{501}(26,\cdot)\) \(\chi_{501}(35,\cdot)\) \(\chi_{501}(41,\cdot)\) \(\chi_{501}(53,\cdot)\) \(\chi_{501}(59,\cdot)\) \(\chi_{501}(68,\cdot)\) \(\chi_{501}(71,\cdot)\) \(\chi_{501}(74,\cdot)\) \(\chi_{501}(80,\cdot)\) \(\chi_{501}(83,\cdot)\) \(\chi_{501}(86,\cdot)\) \(\chi_{501}(92,\cdot)\) \(\chi_{501}(95,\cdot)\) \(\chi_{501}(101,\cdot)\) \(\chi_{501}(104,\cdot)\) \(\chi_{501}(110,\cdot)\) \(\chi_{501}(113,\cdot)\) \(\chi_{501}(119,\cdot)\) \(\chi_{501}(125,\cdot)\) \(\chi_{501}(131,\cdot)\) \(\chi_{501}(134,\cdot)\) \(\chi_{501}(140,\cdot)\) \(\chi_{501}(143,\cdot)\) \(\chi_{501}(146,\cdot)\) \(\chi_{501}(149,\cdot)\) \(\chi_{501}(155,\cdot)\) ...

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

\((335,172)\) → \((-1,e\left(\frac{53}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 501 }(17, a) \) \(1\)\(1\)\(e\left(\frac{45}{166}\right)\)\(e\left(\frac{45}{83}\right)\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{56}{83}\right)\)\(e\left(\frac{135}{166}\right)\)\(e\left(\frac{15}{166}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{147}{166}\right)\)\(e\left(\frac{157}{166}\right)\)\(e\left(\frac{7}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 501 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 501 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 501 }(17,·),\chi_{ 501 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 501 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2