Properties

Label 229.226
Modulus $229$
Conductor $229$
Order $114$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([47]))
 
pari: [g,chi] = znchar(Mod(226,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
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.k

\(\chi_{229}(5,\cdot)\) \(\chi_{229}(12,\cdot)\) \(\chi_{229}(33,\cdot)\) \(\chi_{229}(36,\cdot)\) \(\chi_{229}(45,\cdot)\) \(\chi_{229}(46,\cdot)\) \(\chi_{229}(49,\cdot)\) \(\chi_{229}(56,\cdot)\) \(\chi_{229}(58,\cdot)\) \(\chi_{229}(62,\cdot)\) \(\chi_{229}(70,\cdot)\) \(\chi_{229}(71,\cdot)\) \(\chi_{229}(76,\cdot)\) \(\chi_{229}(78,\cdot)\) \(\chi_{229}(80,\cdot)\) \(\chi_{229}(85,\cdot)\) \(\chi_{229}(97,\cdot)\) \(\chi_{229}(99,\cdot)\) \(\chi_{229}(100,\cdot)\) \(\chi_{229}(103,\cdot)\) \(\chi_{229}(118,\cdot)\) \(\chi_{229}(138,\cdot)\) \(\chi_{229}(146,\cdot)\) \(\chi_{229}(147,\cdot)\) \(\chi_{229}(148,\cdot)\) \(\chi_{229}(154,\cdot)\) \(\chi_{229}(174,\cdot)\) \(\chi_{229}(178,\cdot)\) \(\chi_{229}(181,\cdot)\) \(\chi_{229}(192,\cdot)\) ...

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 114 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{47}{114}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 229 }(226, a) \) \(1\)\(1\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{47}{114}\right)\)\(e\left(\frac{13}{114}\right)\)\(e\left(\frac{37}{38}\right)\)\(e\left(\frac{29}{57}\right)\)\(e\left(\frac{7}{114}\right)\)\(e\left(\frac{15}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 229 }(226,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(226,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(226,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(226,·)) \;\) at \(\; a,b = \) e.g. 1,2