Properties

Label 709.19
Modulus $709$
Conductor $709$
Order $177$
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([163]))
 
pari: [g,chi] = znchar(Mod(19,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(177\)
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.i

\(\chi_{709}(3,\cdot)\) \(\chi_{709}(7,\cdot)\) \(\chi_{709}(9,\cdot)\) \(\chi_{709}(16,\cdot)\) \(\chi_{709}(19,\cdot)\) \(\chi_{709}(21,\cdot)\) \(\chi_{709}(25,\cdot)\) \(\chi_{709}(29,\cdot)\) \(\chi_{709}(46,\cdot)\) \(\chi_{709}(48,\cdot)\) \(\chi_{709}(49,\cdot)\) \(\chi_{709}(55,\cdot)\) \(\chi_{709}(57,\cdot)\) \(\chi_{709}(60,\cdot)\) \(\chi_{709}(62,\cdot)\) \(\chi_{709}(67,\cdot)\) \(\chi_{709}(74,\cdot)\) \(\chi_{709}(81,\cdot)\) \(\chi_{709}(112,\cdot)\) \(\chi_{709}(113,\cdot)\) \(\chi_{709}(121,\cdot)\) \(\chi_{709}(127,\cdot)\) \(\chi_{709}(130,\cdot)\) \(\chi_{709}(132,\cdot)\) \(\chi_{709}(133,\cdot)\) \(\chi_{709}(136,\cdot)\) \(\chi_{709}(140,\cdot)\) \(\chi_{709}(157,\cdot)\) \(\chi_{709}(177,\cdot)\) \(\chi_{709}(180,\cdot)\) ...

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

\(2\) → \(e\left(\frac{163}{177}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{163}{177}\right)\)\(e\left(\frac{70}{177}\right)\)\(e\left(\frac{149}{177}\right)\)\(e\left(\frac{94}{177}\right)\)\(e\left(\frac{56}{177}\right)\)\(e\left(\frac{157}{177}\right)\)\(e\left(\frac{45}{59}\right)\)\(e\left(\frac{140}{177}\right)\)\(e\left(\frac{80}{177}\right)\)\(e\left(\frac{1}{177}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(19,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(19,r) e\left(\frac{2r}{709}\right) = -1.8500449787+-26.5627056901i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(19,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(19,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(19,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(19,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(19,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -40.7331136082+10.3352734357i \)