Properties

Conductor 79
Order 13
Real No
Primitive Yes
Parity Even
Orbit Label 79.e

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(79)
sage: chi = H[46]
pari: [g,chi] = znchar(Mod(46,79))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 79
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 13
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 = 79.e
Orbit index = 5

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 ]

\(\chi_{79}(8,\cdot)\) \(\chi_{79}(10,\cdot)\) \(\chi_{79}(18,\cdot)\) \(\chi_{79}(21,\cdot)\) \(\chi_{79}(22,\cdot)\) \(\chi_{79}(38,\cdot)\) \(\chi_{79}(46,\cdot)\) \(\chi_{79}(52,\cdot)\) \(\chi_{79}(62,\cdot)\) \(\chi_{79}(64,\cdot)\) \(\chi_{79}(65,\cdot)\) \(\chi_{79}(67,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{5}{13}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{2}{13}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{13})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 79 }(46,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{79}(46,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(46,r) e\left(\frac{2r}{79}\right) = -8.1074731407+-3.6426472891i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 79 }(46,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{79}(46,\cdot),\chi_{79}(1,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(46,r) \chi_{79}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 79 }(46,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{79}(46,·)) = \sum_{r \in \Z/79\Z} \chi_{79}(46,r) e\left(\frac{1 r + 2 r^{-1}}{79}\right) = -0.2392919147+1.9707462219i \)