Properties

Label 668.55
Modulus $668$
Conductor $668$
Order $166$
Real no
Primitive yes
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([83,29]))
 
pari: [g,chi] = znchar(Mod(55,668))
 

Basic properties

Modulus: \(668\)
Conductor: \(668\)
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 668.h

\(\chi_{668}(15,\cdot)\) \(\chi_{668}(23,\cdot)\) \(\chi_{668}(35,\cdot)\) \(\chi_{668}(39,\cdot)\) \(\chi_{668}(43,\cdot)\) \(\chi_{668}(51,\cdot)\) \(\chi_{668}(55,\cdot)\) \(\chi_{668}(59,\cdot)\) \(\chi_{668}(67,\cdot)\) \(\chi_{668}(71,\cdot)\) \(\chi_{668}(79,\cdot)\) \(\chi_{668}(83,\cdot)\) \(\chi_{668}(91,\cdot)\) \(\chi_{668}(95,\cdot)\) \(\chi_{668}(103,\cdot)\) \(\chi_{668}(111,\cdot)\) \(\chi_{668}(119,\cdot)\) \(\chi_{668}(123,\cdot)\) \(\chi_{668}(131,\cdot)\) \(\chi_{668}(135,\cdot)\) \(\chi_{668}(139,\cdot)\) \(\chi_{668}(143,\cdot)\) \(\chi_{668}(151,\cdot)\) \(\chi_{668}(155,\cdot)\) \(\chi_{668}(159,\cdot)\) \(\chi_{668}(163,\cdot)\) \(\chi_{668}(187,\cdot)\) \(\chi_{668}(207,\cdot)\) \(\chi_{668}(219,\cdot)\) \(\chi_{668}(227,\cdot)\) ...

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

\((335,5)\) → \((-1,e\left(\frac{29}{166}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{153}{166}\right)\)\(e\left(\frac{29}{166}\right)\)\(e\left(\frac{19}{166}\right)\)\(e\left(\frac{70}{83}\right)\)\(e\left(\frac{65}{166}\right)\)\(e\left(\frac{165}{166}\right)\)\(e\left(\frac{8}{83}\right)\)\(e\left(\frac{43}{166}\right)\)\(e\left(\frac{105}{166}\right)\)\(e\left(\frac{3}{83}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 668 }(55,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{668}(55,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(55,r) e\left(\frac{r}{334}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 668 }(55,·),\chi_{ 668 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{668}(55,\cdot),\chi_{668}(1,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(55,r) \chi_{668}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 668 }(55,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{668}(55,·)) = \sum_{r \in \Z/668\Z} \chi_{668}(55,r) e\left(\frac{1 r + 2 r^{-1}}{668}\right) = 6.8149576641+-0.2580729053i \)