Properties

Label 668.9
Modulus $668$
Conductor $167$
Order $83$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(668)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(9,668))
 

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 \(\chi_{167}(9,\cdot)\)
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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((335,5)\) → \((1,e\left(\frac{11}{83}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{38}{83}\right)\)\(e\left(\frac{11}{83}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{59}{83}\right)\)\(e\left(\frac{54}{83}\right)\)\(e\left(\frac{49}{83}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{57}{83}\right)\)\(e\left(\frac{8}{83}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 668 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{668}(9,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(9,r) e\left(\frac{r}{334}\right) = 14.5883823605+-21.3349267658i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 668 }(9,·),\chi_{ 668 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{668}(9,\cdot),\chi_{668}(1,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(9,r) \chi_{668}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 668 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{668}(9,·)) = \sum_{r \in \Z/668\Z} \chi_{668}(9,r) e\left(\frac{1 r + 2 r^{-1}}{668}\right) = 0.0 \)