Properties

Conductor 79
Order 78
Real No
Primitive Yes
Parity Odd
Orbit Label 79.h

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[68]
pari: [g,chi] = znchar(Mod(68,79))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 79
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 78
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 = Odd
Orbit label = 79.h
Orbit index = 8

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 ]

\(\chi_{79}(3,\cdot)\) \(\chi_{79}(6,\cdot)\) \(\chi_{79}(7,\cdot)\) \(\chi_{79}(28,\cdot)\) \(\chi_{79}(29,\cdot)\) \(\chi_{79}(30,\cdot)\) \(\chi_{79}(34,\cdot)\) \(\chi_{79}(35,\cdot)\) \(\chi_{79}(37,\cdot)\) \(\chi_{79}(39,\cdot)\) \(\chi_{79}(43,\cdot)\) \(\chi_{79}(47,\cdot)\) \(\chi_{79}(48,\cdot)\) \(\chi_{79}(53,\cdot)\) \(\chi_{79}(54,\cdot)\) \(\chi_{79}(59,\cdot)\) \(\chi_{79}(60,\cdot)\) \(\chi_{79}(63,\cdot)\) \(\chi_{79}(66,\cdot)\) \(\chi_{79}(68,\cdot)\) \(\chi_{79}(70,\cdot)\) \(\chi_{79}(74,\cdot)\) \(\chi_{79}(75,\cdot)\) \(\chi_{79}(77,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{29}{78}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{11}{39}\right)\)
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)
\( \tau_{ a }( \chi_{ 79 }(68,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{79}(68,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(68,r) e\left(\frac{2r}{79}\right) = -3.0490260258+-8.3488586222i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 79 }(68,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{79}(68,\cdot),\chi_{79}(1,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(68,r) \chi_{79}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 79 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{79}(68,·)) = \sum_{r \in \Z/79\Z} \chi_{79}(68,r) e\left(\frac{1 r + 2 r^{-1}}{79}\right) = -6.2518032609+0.2519390587i \)