Properties

Label 709.175
Modulus $709$
Conductor $709$
Order $59$
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(709)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([58]))
 
pari: [g,chi] = znchar(Mod(175,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(59\)
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.g

\(\chi_{709}(20,\cdot)\) \(\chi_{709}(27,\cdot)\) \(\chi_{709}(44,\cdot)\) \(\chi_{709}(59,\cdot)\) \(\chi_{709}(63,\cdot)\) \(\chi_{709}(75,\cdot)\) \(\chi_{709}(82,\cdot)\) \(\chi_{709}(87,\cdot)\) \(\chi_{709}(104,\cdot)\) \(\chi_{709}(138,\cdot)\) \(\chi_{709}(144,\cdot)\) \(\chi_{709}(147,\cdot)\) \(\chi_{709}(149,\cdot)\) \(\chi_{709}(163,\cdot)\) \(\chi_{709}(165,\cdot)\) \(\chi_{709}(170,\cdot)\) \(\chi_{709}(171,\cdot)\) \(\chi_{709}(172,\cdot)\) \(\chi_{709}(175,\cdot)\) \(\chi_{709}(181,\cdot)\) \(\chi_{709}(186,\cdot)\) \(\chi_{709}(201,\cdot)\) \(\chi_{709}(203,\cdot)\) \(\chi_{709}(222,\cdot)\) \(\chi_{709}(283,\cdot)\) \(\chi_{709}(322,\cdot)\) \(\chi_{709}(336,\cdot)\) \(\chi_{709}(339,\cdot)\) \(\chi_{709}(343,\cdot)\) \(\chi_{709}(363,\cdot)\) ...

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

\(2\) → \(e\left(\frac{58}{59}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{58}{59}\right)\)\(e\left(\frac{5}{59}\right)\)\(e\left(\frac{57}{59}\right)\)\(e\left(\frac{32}{59}\right)\)\(e\left(\frac{4}{59}\right)\)\(e\left(\frac{7}{59}\right)\)\(e\left(\frac{56}{59}\right)\)\(e\left(\frac{10}{59}\right)\)\(e\left(\frac{31}{59}\right)\)\(e\left(\frac{38}{59}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{59})$
Fixed field: Number field defined by a degree 59 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(175,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(175,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(175,r) e\left(\frac{2r}{709}\right) = 24.4356320171+10.5782743358i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(175,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(175,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(175,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(175,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(175,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(175,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -5.3723077593+0.2863317221i \)