Properties

Label 709.4
Modulus $709$
Conductor $709$
Order $354$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(709, base_ring=CyclotomicField(354))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(4,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(354\)
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 709.k

\(\chi_{709}(4,\cdot)\) \(\chi_{709}(5,\cdot)\) \(\chi_{709}(11,\cdot)\) \(\chi_{709}(15,\cdot)\) \(\chi_{709}(26,\cdot)\) \(\chi_{709}(33,\cdot)\) \(\chi_{709}(34,\cdot)\) \(\chi_{709}(35,\cdot)\) \(\chi_{709}(36,\cdot)\) \(\chi_{709}(43,\cdot)\) \(\chi_{709}(77,\cdot)\) \(\chi_{709}(78,\cdot)\) \(\chi_{709}(80,\cdot)\) \(\chi_{709}(84,\cdot)\) \(\chi_{709}(95,\cdot)\) \(\chi_{709}(100,\cdot)\) \(\chi_{709}(103,\cdot)\) \(\chi_{709}(106,\cdot)\) \(\chi_{709}(108,\cdot)\) \(\chi_{709}(116,\cdot)\) \(\chi_{709}(122,\cdot)\) \(\chi_{709}(129,\cdot)\) \(\chi_{709}(135,\cdot)\) \(\chi_{709}(141,\cdot)\) \(\chi_{709}(142,\cdot)\) \(\chi_{709}(146,\cdot)\) \(\chi_{709}(151,\cdot)\) \(\chi_{709}(166,\cdot)\) \(\chi_{709}(176,\cdot)\) \(\chi_{709}(178,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{177})$
Fixed field: Number field defined by a degree 354 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{354}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 709 }(4, a) \) \(1\)\(1\)\(e\left(\frac{1}{354}\right)\)\(e\left(\frac{86}{177}\right)\)\(e\left(\frac{1}{177}\right)\)\(e\left(\frac{161}{177}\right)\)\(e\left(\frac{173}{354}\right)\)\(e\left(\frac{26}{177}\right)\)\(e\left(\frac{1}{118}\right)\)\(e\left(\frac{172}{177}\right)\)\(e\left(\frac{323}{354}\right)\)\(e\left(\frac{158}{177}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 709 }(4,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(4,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2