Properties

Label 517.13
Modulus $517$
Conductor $517$
Order $230$
Real no
Primitive yes
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(230))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23,55]))
 
pari: [g,chi] = znchar(Mod(13,517))
 

Basic properties

Modulus: \(517\)
Conductor: \(517\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(230\)
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 517.n

\(\chi_{517}(13,\cdot)\) \(\chi_{517}(19,\cdot)\) \(\chi_{517}(29,\cdot)\) \(\chi_{517}(30,\cdot)\) \(\chi_{517}(35,\cdot)\) \(\chi_{517}(39,\cdot)\) \(\chi_{517}(40,\cdot)\) \(\chi_{517}(41,\cdot)\) \(\chi_{517}(52,\cdot)\) \(\chi_{517}(57,\cdot)\) \(\chi_{517}(62,\cdot)\) \(\chi_{517}(73,\cdot)\) \(\chi_{517}(85,\cdot)\) \(\chi_{517}(90,\cdot)\) \(\chi_{517}(105,\cdot)\) \(\chi_{517}(107,\cdot)\) \(\chi_{517}(116,\cdot)\) \(\chi_{517}(117,\cdot)\) \(\chi_{517}(123,\cdot)\) \(\chi_{517}(127,\cdot)\) \(\chi_{517}(129,\cdot)\) \(\chi_{517}(134,\cdot)\) \(\chi_{517}(138,\cdot)\) \(\chi_{517}(139,\cdot)\) \(\chi_{517}(151,\cdot)\) \(\chi_{517}(156,\cdot)\) \(\chi_{517}(160,\cdot)\) \(\chi_{517}(161,\cdot)\) \(\chi_{517}(167,\cdot)\) \(\chi_{517}(171,\cdot)\) ...

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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((189,287)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{11}{46}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(13, a) \) \(1\)\(1\)\(e\left(\frac{93}{230}\right)\)\(e\left(\frac{67}{115}\right)\)\(e\left(\frac{93}{115}\right)\)\(e\left(\frac{147}{230}\right)\)\(e\left(\frac{227}{230}\right)\)\(e\left(\frac{81}{230}\right)\)\(e\left(\frac{49}{230}\right)\)\(e\left(\frac{19}{115}\right)\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{9}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(13,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2