Properties

Label 709.346
Modulus $709$
Conductor $709$
Order $118$
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(118))
 
M = H._module
 
chi = DirichletCharacter(H, M([75]))
 
pari: [g,chi] = znchar(Mod(346,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(118\)
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.h

\(\chi_{709}(12,\cdot)\) \(\chi_{709}(28,\cdot)\) \(\chi_{709}(45,\cdot)\) \(\chi_{709}(47,\cdot)\) \(\chi_{709}(64,\cdot)\) \(\chi_{709}(76,\cdot)\) \(\chi_{709}(99,\cdot)\) \(\chi_{709}(102,\cdot)\) \(\chi_{709}(105,\cdot)\) \(\chi_{709}(125,\cdot)\) \(\chi_{709}(145,\cdot)\) \(\chi_{709}(158,\cdot)\) \(\chi_{709}(169,\cdot)\) \(\chi_{709}(191,\cdot)\) \(\chi_{709}(194,\cdot)\) \(\chi_{709}(230,\cdot)\) \(\chi_{709}(231,\cdot)\) \(\chi_{709}(234,\cdot)\) \(\chi_{709}(238,\cdot)\) \(\chi_{709}(240,\cdot)\) \(\chi_{709}(245,\cdot)\) \(\chi_{709}(275,\cdot)\) \(\chi_{709}(285,\cdot)\) \(\chi_{709}(309,\cdot)\) \(\chi_{709}(310,\cdot)\) \(\chi_{709}(319,\cdot)\) \(\chi_{709}(324,\cdot)\) \(\chi_{709}(335,\cdot)\) \(\chi_{709}(346,\cdot)\) \(\chi_{709}(366,\cdot)\) ...

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: $\Q(\zeta_{59})$
Fixed field: Number field defined by a degree 118 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{75}{118}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 709 }(346, a) \) \(1\)\(1\)\(e\left(\frac{75}{118}\right)\)\(e\left(\frac{19}{59}\right)\)\(e\left(\frac{16}{59}\right)\)\(e\left(\frac{39}{59}\right)\)\(e\left(\frac{113}{118}\right)\)\(e\left(\frac{3}{59}\right)\)\(e\left(\frac{107}{118}\right)\)\(e\left(\frac{38}{59}\right)\)\(e\left(\frac{35}{118}\right)\)\(e\left(\frac{50}{59}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 709 }(346,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(346,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(346,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(346,·)) \;\) at \(\; a,b = \) e.g. 1,2