Properties

Label 633.20
Modulus $633$
Conductor $633$
Order $210$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,134]))
 
pari: [g,chi] = znchar(Mod(20,633))
 

Basic properties

Modulus: \(633\)
Conductor: \(633\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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 633.be

\(\chi_{633}(20,\cdot)\) \(\chi_{633}(44,\cdot)\) \(\chi_{633}(47,\cdot)\) \(\chi_{633}(53,\cdot)\) \(\chi_{633}(56,\cdot)\) \(\chi_{633}(59,\cdot)\) \(\chi_{633}(62,\cdot)\) \(\chi_{633}(80,\cdot)\) \(\chi_{633}(95,\cdot)\) \(\chi_{633}(119,\cdot)\) \(\chi_{633}(170,\cdot)\) \(\chi_{633}(176,\cdot)\) \(\chi_{633}(182,\cdot)\) \(\chi_{633}(194,\cdot)\) \(\chi_{633}(209,\cdot)\) \(\chi_{633}(215,\cdot)\) \(\chi_{633}(227,\cdot)\) \(\chi_{633}(248,\cdot)\) \(\chi_{633}(257,\cdot)\) \(\chi_{633}(260,\cdot)\) \(\chi_{633}(263,\cdot)\) \(\chi_{633}(281,\cdot)\) \(\chi_{633}(314,\cdot)\) \(\chi_{633}(347,\cdot)\) \(\chi_{633}(350,\cdot)\) \(\chi_{633}(365,\cdot)\) \(\chi_{633}(374,\cdot)\) \(\chi_{633}(383,\cdot)\) \(\chi_{633}(419,\cdot)\) \(\chi_{633}(428,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((212,424)\) → \((-1,e\left(\frac{67}{105}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 633 }(20, a) \) \(-1\)\(1\)\(e\left(\frac{29}{210}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{73}{105}\right)\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{58}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 633 }(20,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 633 }(20,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 633 }(20,·),\chi_{ 633 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 633 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2