Properties

Label 79.3
Modulus $79$
Conductor $79$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(79, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(3,79))
 

Basic properties

Modulus: \(79\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 79.h

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

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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 78 polynomial

Values on generators

\(3\) → \(e\left(\frac{1}{78}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 79 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 79 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 79 }(3,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 79 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2