Properties

Label 229.31
Modulus $229$
Conductor $229$
Order $228$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(228))
 
M = H._module
 
chi = DirichletCharacter(H, M([29]))
 
pari: [g,chi] = znchar(Mod(31,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
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 229.l

\(\chi_{229}(6,\cdot)\) \(\chi_{229}(7,\cdot)\) \(\chi_{229}(10,\cdot)\) \(\chi_{229}(23,\cdot)\) \(\chi_{229}(24,\cdot)\) \(\chi_{229}(28,\cdot)\) \(\chi_{229}(29,\cdot)\) \(\chi_{229}(31,\cdot)\) \(\chi_{229}(35,\cdot)\) \(\chi_{229}(38,\cdot)\) \(\chi_{229}(39,\cdot)\) \(\chi_{229}(40,\cdot)\) \(\chi_{229}(41,\cdot)\) \(\chi_{229}(47,\cdot)\) \(\chi_{229}(50,\cdot)\) \(\chi_{229}(59,\cdot)\) \(\chi_{229}(63,\cdot)\) \(\chi_{229}(65,\cdot)\) \(\chi_{229}(66,\cdot)\) \(\chi_{229}(67,\cdot)\) \(\chi_{229}(69,\cdot)\) \(\chi_{229}(72,\cdot)\) \(\chi_{229}(73,\cdot)\) \(\chi_{229}(74,\cdot)\) \(\chi_{229}(77,\cdot)\) \(\chi_{229}(79,\cdot)\) \(\chi_{229}(87,\cdot)\) \(\chi_{229}(90,\cdot)\) \(\chi_{229}(92,\cdot)\) \(\chi_{229}(96,\cdot)\) ...

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: $\Q(\zeta_{228})$
Fixed field: Number field defined by a degree 228 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{29}{228}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 229 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{51}{76}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{13}{38}\right)\)\(e\left(\frac{53}{114}\right)\)\(e\left(\frac{29}{228}\right)\)\(e\left(\frac{139}{228}\right)\)\(e\left(\frac{1}{76}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{31}{228}\right)\)\(e\left(\frac{23}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 229 }(31,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(31,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(31,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2