Properties

Label 709.14
Modulus $709$
Conductor $709$
Order $708$
Real no
Primitive yes
Minimal yes
Parity odd

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([53]))
 
pari: [g,chi] = znchar(Mod(14,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(708\)
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 709.l

\(\chi_{709}(2,\cdot)\) \(\chi_{709}(6,\cdot)\) \(\chi_{709}(10,\cdot)\) \(\chi_{709}(14,\cdot)\) \(\chi_{709}(17,\cdot)\) \(\chi_{709}(22,\cdot)\) \(\chi_{709}(23,\cdot)\) \(\chi_{709}(24,\cdot)\) \(\chi_{709}(31,\cdot)\) \(\chi_{709}(32,\cdot)\) \(\chi_{709}(37,\cdot)\) \(\chi_{709}(38,\cdot)\) \(\chi_{709}(39,\cdot)\) \(\chi_{709}(40,\cdot)\) \(\chi_{709}(41,\cdot)\) \(\chi_{709}(51,\cdot)\) \(\chi_{709}(52,\cdot)\) \(\chi_{709}(54,\cdot)\) \(\chi_{709}(56,\cdot)\) \(\chi_{709}(61,\cdot)\) \(\chi_{709}(65,\cdot)\) \(\chi_{709}(69,\cdot)\) \(\chi_{709}(71,\cdot)\) \(\chi_{709}(72,\cdot)\) \(\chi_{709}(79,\cdot)\) \(\chi_{709}(85,\cdot)\) \(\chi_{709}(86,\cdot)\) \(\chi_{709}(88,\cdot)\) \(\chi_{709}(89,\cdot)\) \(\chi_{709}(90,\cdot)\) ...

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

\(2\) → \(e\left(\frac{53}{708}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{53}{708}\right)\)\(e\left(\frac{155}{177}\right)\)\(e\left(\frac{53}{354}\right)\)\(e\left(\frac{37}{354}\right)\)\(e\left(\frac{673}{708}\right)\)\(e\left(\frac{158}{177}\right)\)\(e\left(\frac{53}{236}\right)\)\(e\left(\frac{133}{177}\right)\)\(e\left(\frac{127}{708}\right)\)\(e\left(\frac{55}{354}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{708})$
Fixed field: Number field defined by a degree 708 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(14,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(14,r) e\left(\frac{2r}{709}\right) = -1.0613734188+26.6058919502i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(14,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(14,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(14,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(14,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(14,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -3.5155565998+14.6720247537i \)