Properties

Label 667.2
Modulus $667$
Conductor $667$
Order $308$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(308))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,11]))
 
pari: [g,chi] = znchar(Mod(2,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
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 667.w

\(\chi_{667}(2,\cdot)\) \(\chi_{667}(3,\cdot)\) \(\chi_{667}(8,\cdot)\) \(\chi_{667}(18,\cdot)\) \(\chi_{667}(26,\cdot)\) \(\chi_{667}(27,\cdot)\) \(\chi_{667}(31,\cdot)\) \(\chi_{667}(32,\cdot)\) \(\chi_{667}(39,\cdot)\) \(\chi_{667}(48,\cdot)\) \(\chi_{667}(50,\cdot)\) \(\chi_{667}(55,\cdot)\) \(\chi_{667}(72,\cdot)\) \(\chi_{667}(73,\cdot)\) \(\chi_{667}(77,\cdot)\) \(\chi_{667}(85,\cdot)\) \(\chi_{667}(95,\cdot)\) \(\chi_{667}(98,\cdot)\) \(\chi_{667}(101,\cdot)\) \(\chi_{667}(105,\cdot)\) \(\chi_{667}(108,\cdot)\) \(\chi_{667}(118,\cdot)\) \(\chi_{667}(119,\cdot)\) \(\chi_{667}(124,\cdot)\) \(\chi_{667}(127,\cdot)\) \(\chi_{667}(131,\cdot)\) \(\chi_{667}(142,\cdot)\) \(\chi_{667}(147,\cdot)\) \(\chi_{667}(156,\cdot)\) \(\chi_{667}(163,\cdot)\) ...

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: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial (not computed)

Values on generators

\((465,553)\) → \((e\left(\frac{1}{11}\right),e\left(\frac{1}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 667 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{67}{308}\right)\)\(e\left(\frac{195}{308}\right)\)\(e\left(\frac{67}{154}\right)\)\(e\left(\frac{135}{154}\right)\)\(e\left(\frac{131}{154}\right)\)\(e\left(\frac{12}{77}\right)\)\(e\left(\frac{201}{308}\right)\)\(e\left(\frac{41}{154}\right)\)\(e\left(\frac{29}{308}\right)\)\(e\left(\frac{219}{308}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 667 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(2,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2