Properties

Label 229.64
Modulus $229$
Conductor $229$
Order $38$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([21]))
 
pari: [g,chi] = znchar(Mod(64,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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 229.h

\(\chi_{229}(4,\cdot)\) \(\chi_{229}(11,\cdot)\) \(\chi_{229}(15,\cdot)\) \(\chi_{229}(26,\cdot)\) \(\chi_{229}(64,\cdot)\) \(\chi_{229}(68,\cdot)\) \(\chi_{229}(108,\cdot)\) \(\chi_{229}(125,\cdot)\) \(\chi_{229}(168,\cdot)\) \(\chi_{229}(169,\cdot)\) \(\chi_{229}(172,\cdot)\) \(\chi_{229}(176,\cdot)\) \(\chi_{229}(185,\cdot)\) \(\chi_{229}(186,\cdot)\) \(\chi_{229}(187,\cdot)\) \(\chi_{229}(202,\cdot)\) \(\chi_{229}(212,\cdot)\) \(\chi_{229}(213,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: Number field defined by a degree 38 polynomial

Values on generators

\(6\) → \(e\left(\frac{21}{38}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 229 }(64, a) \) \(1\)\(1\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{5}{38}\right)\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{10}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 229 }(64,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(64,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(64,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2