Properties

Label 829.697
Modulus $829$
Conductor $829$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(829, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([36]))
 
pari: [g,chi] = znchar(Mod(697,829))
 

Basic properties

Modulus: \(829\)
Conductor: \(829\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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 829.i

\(\chi_{829}(11,\cdot)\) \(\chi_{829}(15,\cdot)\) \(\chi_{829}(56,\cdot)\) \(\chi_{829}(59,\cdot)\) \(\chi_{829}(69,\cdot)\) \(\chi_{829}(121,\cdot)\) \(\chi_{829}(144,\cdot)\) \(\chi_{829}(157,\cdot)\) \(\chi_{829}(165,\cdot)\) \(\chi_{829}(206,\cdot)\) \(\chi_{829}(225,\cdot)\) \(\chi_{829}(502,\cdot)\) \(\chi_{829}(507,\cdot)\) \(\chi_{829}(548,\cdot)\) \(\chi_{829}(603,\cdot)\) \(\chi_{829}(608,\cdot)\) \(\chi_{829}(616,\cdot)\) \(\chi_{829}(649,\cdot)\) \(\chi_{829}(697,\cdot)\) \(\chi_{829}(755,\cdot)\) \(\chi_{829}(759,\cdot)\) \(\chi_{829}(817,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(2\) → \(e\left(\frac{18}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 829 }(697, a) \) \(1\)\(1\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(1\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{7}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 829 }(697,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 829 }(697,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 829 }(697,·),\chi_{ 829 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 829 }(697,·)) \;\) at \(\; a,b = \) e.g. 1,2