Properties

Label 517.34
Modulus $517$
Conductor $47$
Order $23$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(517, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,34]))
 
pari: [g,chi] = znchar(Mod(34,517))
 

Basic properties

Modulus: \(517\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{47}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 517.i

\(\chi_{517}(12,\cdot)\) \(\chi_{517}(34,\cdot)\) \(\chi_{517}(56,\cdot)\) \(\chi_{517}(89,\cdot)\) \(\chi_{517}(100,\cdot)\) \(\chi_{517}(111,\cdot)\) \(\chi_{517}(122,\cdot)\) \(\chi_{517}(144,\cdot)\) \(\chi_{517}(155,\cdot)\) \(\chi_{517}(166,\cdot)\) \(\chi_{517}(177,\cdot)\) \(\chi_{517}(243,\cdot)\) \(\chi_{517}(298,\cdot)\) \(\chi_{517}(309,\cdot)\) \(\chi_{517}(331,\cdot)\) \(\chi_{517}(353,\cdot)\) \(\chi_{517}(397,\cdot)\) \(\chi_{517}(408,\cdot)\) \(\chi_{517}(430,\cdot)\) \(\chi_{517}(441,\cdot)\) \(\chi_{517}(474,\cdot)\) \(\chi_{517}(507,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{23})\)
Fixed field: \(\Q(\zeta_{47})^+\)

Values on generators

\((189,287)\) → \((1,e\left(\frac{17}{23}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(34, a) \) \(1\)\(1\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{9}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(34,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(34,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(34,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2