Properties

Modulus 709
Conductor 709
Order 354
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 709.k

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([247]))
 
pari: [g,chi] = znchar(Mod(15,709))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 709
Conductor = 709
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 354
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 709.k
Orbit index = 11

Galois orbit

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

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

Values on generators

\(2\) → \(e\left(\frac{247}{354}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{247}{354}\right)\)\(e\left(\frac{2}{177}\right)\)\(e\left(\frac{70}{177}\right)\)\(e\left(\frac{119}{177}\right)\)\(e\left(\frac{251}{354}\right)\)\(e\left(\frac{50}{177}\right)\)\(e\left(\frac{11}{118}\right)\)\(e\left(\frac{4}{177}\right)\)\(e\left(\frac{131}{354}\right)\)\(e\left(\frac{86}{177}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{177})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(15,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(15,r) e\left(\frac{2r}{709}\right) = 5.2082885045+-26.1127120547i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(15,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(15,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(15,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(15,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(15,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -18.1025787844+25.2917366915i \)