Properties

Label 79.65
Modulus $79$
Conductor $79$
Order $13$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(79)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(65,79))
 

Basic properties

Modulus: \(79\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 79.e

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

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

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{9}{13}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: 13.13.59091511031674153381441.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 79 }(65,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{79}(65,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(65,r) e\left(\frac{2r}{79}\right) = 5.2268501542+7.1888829081i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 79 }(65,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{79}(65,\cdot),\chi_{79}(1,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(65,r) \chi_{79}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 79 }(65,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{79}(65,·)) = \sum_{r \in \Z/79\Z} \chi_{79}(65,r) e\left(\frac{1 r + 2 r^{-1}}{79}\right) = -5.5623657742+1.3710000294i \)