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[41]
pari: [g,chi] = znchar(Mod(41,131))

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

\(2\) → \(e\left(\frac{63}{65}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{51}{65}\right)\)\(e\left(\frac{61}{65}\right)\)\(e\left(\frac{38}{65}\right)\)\(e\left(\frac{49}{65}\right)\)\(e\left(\frac{3}{65}\right)\)\(e\left(\frac{59}{65}\right)\)\(e\left(\frac{37}{65}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{18}{65}\right)\)
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)
\( \tau_{ a }( \chi_{ 131 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{131}(41,\cdot)) = \sum_{r\in \Z/131\Z} \chi_{131}(41,r) e\left(\frac{2r}{131}\right) = 4.3427534761+-10.5896407987i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 131 }(41,·),\chi_{ 131 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{131}(41,\cdot),\chi_{131}(1,\cdot)) = \sum_{r\in \Z/131\Z} \chi_{131}(41,r) \chi_{131}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 131 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{131}(41,·)) = \sum_{r \in \Z/131\Z} \chi_{131}(41,r) e\left(\frac{1 r + 2 r^{-1}}{131}\right) = -6.1315583325+0.5945563422i \)