Properties

Conductor 79
Order 39
Real No
Primitive Yes
Parity Even
Orbit Label 79.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(79)
sage: chi = H[26]
pari: [g,chi] = znchar(Mod(26,79))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 79
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 39
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.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 ]

\(\chi_{79}(2,\cdot)\) \(\chi_{79}(4,\cdot)\) \(\chi_{79}(5,\cdot)\) \(\chi_{79}(9,\cdot)\) \(\chi_{79}(11,\cdot)\) \(\chi_{79}(13,\cdot)\) \(\chi_{79}(16,\cdot)\) \(\chi_{79}(19,\cdot)\) \(\chi_{79}(20,\cdot)\) \(\chi_{79}(25,\cdot)\) \(\chi_{79}(26,\cdot)\) \(\chi_{79}(31,\cdot)\) \(\chi_{79}(32,\cdot)\) \(\chi_{79}(36,\cdot)\) \(\chi_{79}(40,\cdot)\) \(\chi_{79}(42,\cdot)\) \(\chi_{79}(44,\cdot)\) \(\chi_{79}(45,\cdot)\) \(\chi_{79}(49,\cdot)\) \(\chi_{79}(50,\cdot)\) \(\chi_{79}(51,\cdot)\) \(\chi_{79}(72,\cdot)\) \(\chi_{79}(73,\cdot)\) \(\chi_{79}(76,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{19}{39}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{5}{39}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{39})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 79 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{79}(26,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(26,r) e\left(\frac{2r}{79}\right) = 8.8874575038+-0.1144513822i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 79 }(26,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{79}(26,\cdot),\chi_{79}(1,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(26,r) \chi_{79}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 79 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{79}(26,·)) = \sum_{r \in \Z/79\Z} \chi_{79}(26,r) e\left(\frac{1 r + 2 r^{-1}}{79}\right) = -3.8080423968+0.6188671252i \)