Properties

Conductor 131
Order 65
Real No
Primitive Yes
Parity Even
Orbit Label 131.g

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(131)
 
sage: chi = H[34]
 
pari: [g,chi] = znchar(Mod(34,131))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 131
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 65
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 131.g
Orbit index = 7

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{131}(3,\cdot)\) \(\chi_{131}(4,\cdot)\) \(\chi_{131}(5,\cdot)\) \(\chi_{131}(7,\cdot)\) \(\chi_{131}(9,\cdot)\) \(\chi_{131}(11,\cdot)\) \(\chi_{131}(12,\cdot)\) \(\chi_{131}(13,\cdot)\) \(\chi_{131}(15,\cdot)\) \(\chi_{131}(16,\cdot)\) \(\chi_{131}(20,\cdot)\) \(\chi_{131}(21,\cdot)\) \(\chi_{131}(25,\cdot)\) \(\chi_{131}(27,\cdot)\) \(\chi_{131}(28,\cdot)\) \(\chi_{131}(33,\cdot)\) \(\chi_{131}(34,\cdot)\) \(\chi_{131}(35,\cdot)\) \(\chi_{131}(36,\cdot)\) \(\chi_{131}(38,\cdot)\) \(\chi_{131}(41,\cdot)\) \(\chi_{131}(43,\cdot)\) \(\chi_{131}(44,\cdot)\) \(\chi_{131}(46,\cdot)\) \(\chi_{131}(48,\cdot)\) \(\chi_{131}(49,\cdot)\) \(\chi_{131}(55,\cdot)\) \(\chi_{131}(59,\cdot)\) \(\chi_{131}(64,\cdot)\) \(\chi_{131}(65,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{22}{65}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{22}{65}\right)\)\(e\left(\frac{24}{65}\right)\)\(e\left(\frac{44}{65}\right)\)\(e\left(\frac{37}{65}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{1}{65}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{59}{65}\right)\)\(e\left(\frac{62}{65}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{65})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 131 }(34,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{131}(34,\cdot)) = \sum_{r\in \Z/131\Z} \chi_{131}(34,r) e\left(\frac{2r}{131}\right) = -4.9180091229+-10.3350465053i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 131 }(34,·),\chi_{ 131 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{131}(34,\cdot),\chi_{131}(1,\cdot)) = \sum_{r\in \Z/131\Z} \chi_{131}(34,r) \chi_{131}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 131 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{131}(34,·)) = \sum_{r \in \Z/131\Z} \chi_{131}(34,r) e\left(\frac{1 r + 2 r^{-1}}{131}\right) = 8.4855669568+15.2600151711i \)