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Properties

Label 79.16
Modulus $79$
Conductor $79$
Order $39$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(79, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([16]))
 
Copy content pari:[g,chi] = znchar(Mod(16,79))
 

Basic properties

Modulus: \(79\)
Conductor: \(79\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(39\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 79.g

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

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

Related number fields

Field of values: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\(3\) → \(e\left(\frac{8}{39}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 79 }(16, a) \) \(1\)\(1\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{37}{39}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 79 }(16,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 79 }(16,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 79 }(16,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 79 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2