Properties

Label 229.111
Modulus $229$
Conductor $229$
Order $57$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([100]))
 
pari: [g,chi] = znchar(Mod(111,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
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.i

\(\chi_{229}(3,\cdot)\) \(\chi_{229}(9,\cdot)\) \(\chi_{229}(14,\cdot)\) \(\chi_{229}(19,\cdot)\) \(\chi_{229}(20,\cdot)\) \(\chi_{229}(25,\cdot)\) \(\chi_{229}(37,\cdot)\) \(\chi_{229}(48,\cdot)\) \(\chi_{229}(51,\cdot)\) \(\chi_{229}(55,\cdot)\) \(\chi_{229}(75,\cdot)\) \(\chi_{229}(81,\cdot)\) \(\chi_{229}(82,\cdot)\) \(\chi_{229}(83,\cdot)\) \(\chi_{229}(91,\cdot)\) \(\chi_{229}(111,\cdot)\) \(\chi_{229}(126,\cdot)\) \(\chi_{229}(129,\cdot)\) \(\chi_{229}(130,\cdot)\) \(\chi_{229}(132,\cdot)\) \(\chi_{229}(144,\cdot)\) \(\chi_{229}(149,\cdot)\) \(\chi_{229}(151,\cdot)\) \(\chi_{229}(153,\cdot)\) \(\chi_{229}(158,\cdot)\) \(\chi_{229}(159,\cdot)\) \(\chi_{229}(167,\cdot)\) \(\chi_{229}(171,\cdot)\) \(\chi_{229}(173,\cdot)\) \(\chi_{229}(180,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\(6\) → \(e\left(\frac{50}{57}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 229 }(111, a) \) \(1\)\(1\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{55}{57}\right)\)\(e\left(\frac{50}{57}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{22}{57}\right)\)\(e\left(\frac{2}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 229 }(111,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(111,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(111,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(111,·)) \;\) at \(\; a,b = \) e.g. 1,2