Properties

Label 667.5
Modulus $667$
Conductor $667$
Order $154$
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(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,121]))
 
pari: [g,chi] = znchar(Mod(5,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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.t

\(\chi_{667}(5,\cdot)\) \(\chi_{667}(33,\cdot)\) \(\chi_{667}(34,\cdot)\) \(\chi_{667}(38,\cdot)\) \(\chi_{667}(42,\cdot)\) \(\chi_{667}(51,\cdot)\) \(\chi_{667}(63,\cdot)\) \(\chi_{667}(67,\cdot)\) \(\chi_{667}(80,\cdot)\) \(\chi_{667}(109,\cdot)\) \(\chi_{667}(120,\cdot)\) \(\chi_{667}(122,\cdot)\) \(\chi_{667}(125,\cdot)\) \(\chi_{667}(129,\cdot)\) \(\chi_{667}(149,\cdot)\) \(\chi_{667}(158,\cdot)\) \(\chi_{667}(178,\cdot)\) \(\chi_{667}(180,\cdot)\) \(\chi_{667}(212,\cdot)\) \(\chi_{667}(237,\cdot)\) \(\chi_{667}(241,\cdot)\) \(\chi_{667}(245,\cdot)\) \(\chi_{667}(267,\cdot)\) \(\chi_{667}(270,\cdot)\) \(\chi_{667}(274,\cdot)\) \(\chi_{667}(283,\cdot)\) \(\chi_{667}(295,\cdot)\) \(\chi_{667}(296,\cdot)\) \(\chi_{667}(332,\cdot)\) \(\chi_{667}(341,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((465,553)\) → \((e\left(\frac{1}{22}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 667 }(5, a) \) \(-1\)\(1\)\(e\left(\frac{135}{154}\right)\)\(e\left(\frac{101}{154}\right)\)\(e\left(\frac{58}{77}\right)\)\(e\left(\frac{51}{154}\right)\)\(e\left(\frac{41}{77}\right)\)\(e\left(\frac{45}{154}\right)\)\(e\left(\frac{97}{154}\right)\)\(e\left(\frac{24}{77}\right)\)\(e\left(\frac{16}{77}\right)\)\(e\left(\frac{4}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 667 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(5,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2