Properties

Label 668.217
Modulus $668$
Conductor $167$
Order $83$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(668, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,42]))
 
pari: [g,chi] = znchar(Mod(217,668))
 

Basic properties

Modulus: \(668\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{167}(50,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 668.e

\(\chi_{668}(9,\cdot)\) \(\chi_{668}(21,\cdot)\) \(\chi_{668}(25,\cdot)\) \(\chi_{668}(29,\cdot)\) \(\chi_{668}(33,\cdot)\) \(\chi_{668}(49,\cdot)\) \(\chi_{668}(57,\cdot)\) \(\chi_{668}(61,\cdot)\) \(\chi_{668}(65,\cdot)\) \(\chi_{668}(77,\cdot)\) \(\chi_{668}(81,\cdot)\) \(\chi_{668}(85,\cdot)\) \(\chi_{668}(89,\cdot)\) \(\chi_{668}(93,\cdot)\) \(\chi_{668}(97,\cdot)\) \(\chi_{668}(121,\cdot)\) \(\chi_{668}(133,\cdot)\) \(\chi_{668}(137,\cdot)\) \(\chi_{668}(141,\cdot)\) \(\chi_{668}(157,\cdot)\) \(\chi_{668}(169,\cdot)\) \(\chi_{668}(173,\cdot)\) \(\chi_{668}(181,\cdot)\) \(\chi_{668}(185,\cdot)\) \(\chi_{668}(189,\cdot)\) \(\chi_{668}(205,\cdot)\) \(\chi_{668}(209,\cdot)\) \(\chi_{668}(217,\cdot)\) \(\chi_{668}(221,\cdot)\) \(\chi_{668}(225,\cdot)\) ...

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

\((335,5)\) → \((1,e\left(\frac{21}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 668 }(217, a) \) \(1\)\(1\)\(e\left(\frac{65}{83}\right)\)\(e\left(\frac{21}{83}\right)\)\(e\left(\frac{71}{83}\right)\)\(e\left(\frac{47}{83}\right)\)\(e\left(\frac{7}{83}\right)\)\(e\left(\frac{5}{83}\right)\)\(e\left(\frac{3}{83}\right)\)\(e\left(\frac{34}{83}\right)\)\(e\left(\frac{56}{83}\right)\)\(e\left(\frac{53}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 668 }(217,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 668 }(217,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 668 }(217,·),\chi_{ 668 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 668 }(217,·)) \;\) at \(\; a,b = \) e.g. 1,2