Properties

Label 517.26
Modulus $517$
Conductor $517$
Order $230$
Real no
Primitive yes
Minimal yes
Parity odd

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([46,145]))
 
pari: [g,chi] = znchar(Mod(26,517))
 

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.p

\(\chi_{517}(5,\cdot)\) \(\chi_{517}(15,\cdot)\) \(\chi_{517}(20,\cdot)\) \(\chi_{517}(26,\cdot)\) \(\chi_{517}(31,\cdot)\) \(\chi_{517}(38,\cdot)\) \(\chi_{517}(58,\cdot)\) \(\chi_{517}(60,\cdot)\) \(\chi_{517}(69,\cdot)\) \(\chi_{517}(70,\cdot)\) \(\chi_{517}(80,\cdot)\) \(\chi_{517}(82,\cdot)\) \(\chi_{517}(86,\cdot)\) \(\chi_{517}(91,\cdot)\) \(\chi_{517}(92,\cdot)\) \(\chi_{517}(104,\cdot)\) \(\chi_{517}(113,\cdot)\) \(\chi_{517}(114,\cdot)\) \(\chi_{517}(124,\cdot)\) \(\chi_{517}(125,\cdot)\) \(\chi_{517}(135,\cdot)\) \(\chi_{517}(137,\cdot)\) \(\chi_{517}(146,\cdot)\) \(\chi_{517}(152,\cdot)\) \(\chi_{517}(163,\cdot)\) \(\chi_{517}(170,\cdot)\) \(\chi_{517}(174,\cdot)\) \(\chi_{517}(179,\cdot)\) \(\chi_{517}(180,\cdot)\) \(\chi_{517}(181,\cdot)\) ...

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

\((189,287)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{29}{46}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{63}{115}\right)\)\(e\left(\frac{24}{115}\right)\)\(e\left(\frac{11}{115}\right)\)\(e\left(\frac{99}{230}\right)\)\(e\left(\frac{87}{115}\right)\)\(e\left(\frac{66}{115}\right)\)\(e\left(\frac{74}{115}\right)\)\(e\left(\frac{48}{115}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{7}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(26,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(26,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(26,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2