Properties

Label 517.162
Modulus $517$
Conductor $517$
Order $230$
Real no
Primitive yes
Minimal yes
Parity odd

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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([69,30]))
 
pari: [g,chi] = znchar(Mod(162,517))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 517.o

\(\chi_{517}(2,\cdot)\) \(\chi_{517}(6,\cdot)\) \(\chi_{517}(7,\cdot)\) \(\chi_{517}(8,\cdot)\) \(\chi_{517}(17,\cdot)\) \(\chi_{517}(18,\cdot)\) \(\chi_{517}(24,\cdot)\) \(\chi_{517}(28,\cdot)\) \(\chi_{517}(50,\cdot)\) \(\chi_{517}(51,\cdot)\) \(\chi_{517}(61,\cdot)\) \(\chi_{517}(63,\cdot)\) \(\chi_{517}(68,\cdot)\) \(\chi_{517}(72,\cdot)\) \(\chi_{517}(74,\cdot)\) \(\chi_{517}(79,\cdot)\) \(\chi_{517}(83,\cdot)\) \(\chi_{517}(84,\cdot)\) \(\chi_{517}(96,\cdot)\) \(\chi_{517}(101,\cdot)\) \(\chi_{517}(106,\cdot)\) \(\chi_{517}(112,\cdot)\) \(\chi_{517}(118,\cdot)\) \(\chi_{517}(128,\cdot)\) \(\chi_{517}(145,\cdot)\) \(\chi_{517}(149,\cdot)\) \(\chi_{517}(150,\cdot)\) \(\chi_{517}(162,\cdot)\) \(\chi_{517}(173,\cdot)\) \(\chi_{517}(178,\cdot)\) ...

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

\((189,287)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{3}{23}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(162, a) \) \(-1\)\(1\)\(e\left(\frac{149}{230}\right)\)\(e\left(\frac{1}{115}\right)\)\(e\left(\frac{34}{115}\right)\)\(e\left(\frac{38}{115}\right)\)\(e\left(\frac{151}{230}\right)\)\(e\left(\frac{63}{230}\right)\)\(e\left(\frac{217}{230}\right)\)\(e\left(\frac{2}{115}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{7}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(162,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(162,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(162,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(162,·)) \;\) at \(\; a,b = \) e.g. 1,2