Properties

Label 647.16
Modulus $647$
Conductor $647$
Order $323$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(647, base_ring=CyclotomicField(646))
 
M = H._module
 
chi = DirichletCharacter(H, M([632]))
 
pari: [g,chi] = znchar(Mod(16,647))
 

Basic properties

Modulus: \(647\)
Conductor: \(647\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(323\)
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 647.g

\(\chi_{647}(2,\cdot)\) \(\chi_{647}(3,\cdot)\) \(\chi_{647}(4,\cdot)\) \(\chi_{647}(6,\cdot)\) \(\chi_{647}(7,\cdot)\) \(\chi_{647}(8,\cdot)\) \(\chi_{647}(9,\cdot)\) \(\chi_{647}(12,\cdot)\) \(\chi_{647}(13,\cdot)\) \(\chi_{647}(14,\cdot)\) \(\chi_{647}(16,\cdot)\) \(\chi_{647}(17,\cdot)\) \(\chi_{647}(18,\cdot)\) \(\chi_{647}(21,\cdot)\) \(\chi_{647}(24,\cdot)\) \(\chi_{647}(25,\cdot)\) \(\chi_{647}(26,\cdot)\) \(\chi_{647}(27,\cdot)\) \(\chi_{647}(28,\cdot)\) \(\chi_{647}(29,\cdot)\) \(\chi_{647}(31,\cdot)\) \(\chi_{647}(32,\cdot)\) \(\chi_{647}(34,\cdot)\) \(\chi_{647}(36,\cdot)\) \(\chi_{647}(39,\cdot)\) \(\chi_{647}(41,\cdot)\) \(\chi_{647}(42,\cdot)\) \(\chi_{647}(48,\cdot)\) \(\chi_{647}(49,\cdot)\) \(\chi_{647}(50,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{323})$
Fixed field: Number field defined by a degree 323 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{316}{323}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 647 }(16, a) \) \(1\)\(1\)\(e\left(\frac{186}{323}\right)\)\(e\left(\frac{22}{323}\right)\)\(e\left(\frac{49}{323}\right)\)\(e\left(\frac{316}{323}\right)\)\(e\left(\frac{208}{323}\right)\)\(e\left(\frac{3}{323}\right)\)\(e\left(\frac{235}{323}\right)\)\(e\left(\frac{44}{323}\right)\)\(e\left(\frac{179}{323}\right)\)\(e\left(\frac{109}{323}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 647 }(16,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 647 }(16,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 647 }(16,·),\chi_{ 647 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 647 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2