Properties

Label 709.26
Modulus $709$
Conductor $709$
Order $354$
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([299]))
 
pari: [g,chi] = znchar(Mod(26,709))
 

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

Values on generators

\(2\) → \(e\left(\frac{299}{354}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{299}{354}\right)\)\(e\left(\frac{49}{177}\right)\)\(e\left(\frac{122}{177}\right)\)\(e\left(\frac{172}{177}\right)\)\(e\left(\frac{43}{354}\right)\)\(e\left(\frac{163}{177}\right)\)\(e\left(\frac{63}{118}\right)\)\(e\left(\frac{98}{177}\right)\)\(e\left(\frac{289}{354}\right)\)\(e\left(\frac{160}{177}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{177})$
Fixed field: Number field defined by a degree 354 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(26,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(26,r) e\left(\frac{2r}{709}\right) = 26.2487235839+-4.472640184i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(26,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(26,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(26,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(26,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(26,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = 23.7760947224+-12.6239357667i \)