Properties

Label 709.30
Modulus $709$
Conductor $709$
Order $236$
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([165]))
 
pari: [g,chi] = znchar(Mod(30,709))
 

Basic properties

Modulus: \(709\)
Conductor: \(709\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(236\)
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.j

\(\chi_{709}(8,\cdot)\) \(\chi_{709}(13,\cdot)\) \(\chi_{709}(18,\cdot)\) \(\chi_{709}(30,\cdot)\) \(\chi_{709}(42,\cdot)\) \(\chi_{709}(50,\cdot)\) \(\chi_{709}(53,\cdot)\) \(\chi_{709}(58,\cdot)\) \(\chi_{709}(66,\cdot)\) \(\chi_{709}(68,\cdot)\) \(\chi_{709}(70,\cdot)\) \(\chi_{709}(73,\cdot)\) \(\chi_{709}(83,\cdot)\) \(\chi_{709}(92,\cdot)\) \(\chi_{709}(98,\cdot)\) \(\chi_{709}(101,\cdot)\) \(\chi_{709}(107,\cdot)\) \(\chi_{709}(109,\cdot)\) \(\chi_{709}(110,\cdot)\) \(\chi_{709}(114,\cdot)\) \(\chi_{709}(123,\cdot)\) \(\chi_{709}(124,\cdot)\) \(\chi_{709}(131,\cdot)\) \(\chi_{709}(134,\cdot)\) \(\chi_{709}(137,\cdot)\) \(\chi_{709}(148,\cdot)\) \(\chi_{709}(153,\cdot)\) \(\chi_{709}(154,\cdot)\) \(\chi_{709}(156,\cdot)\) \(\chi_{709}(160,\cdot)\) ...

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

\(2\) → \(e\left(\frac{165}{236}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{165}{236}\right)\)\(e\left(\frac{15}{59}\right)\)\(e\left(\frac{47}{118}\right)\)\(e\left(\frac{15}{118}\right)\)\(e\left(\frac{225}{236}\right)\)\(e\left(\frac{21}{59}\right)\)\(e\left(\frac{23}{236}\right)\)\(e\left(\frac{30}{59}\right)\)\(e\left(\frac{195}{236}\right)\)\(e\left(\frac{51}{118}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{236})$
Fixed field: Number field defined by a degree 236 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 709 }(30,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(30,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(30,r) e\left(\frac{2r}{709}\right) = -3.1906090261+26.4352040666i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 709 }(30,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(30,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(30,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 709 }(30,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(30,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(30,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -20.9371362928+-15.1267174456i \)