Properties

Label 229.123
Modulus $229$
Conductor $229$
Order $76$
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(76))
 
M = H._module
 
chi = DirichletCharacter(H, M([13]))
 
pari: [g,chi] = znchar(Mod(123,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(76\)
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.j

\(\chi_{229}(2,\cdot)\) \(\chi_{229}(8,\cdot)\) \(\chi_{229}(13,\cdot)\) \(\chi_{229}(21,\cdot)\) \(\chi_{229}(22,\cdot)\) \(\chi_{229}(30,\cdot)\) \(\chi_{229}(32,\cdot)\) \(\chi_{229}(34,\cdot)\) \(\chi_{229}(52,\cdot)\) \(\chi_{229}(54,\cdot)\) \(\chi_{229}(84,\cdot)\) \(\chi_{229}(86,\cdot)\) \(\chi_{229}(88,\cdot)\) \(\chi_{229}(93,\cdot)\) \(\chi_{229}(101,\cdot)\) \(\chi_{229}(106,\cdot)\) \(\chi_{229}(109,\cdot)\) \(\chi_{229}(114,\cdot)\) \(\chi_{229}(115,\cdot)\) \(\chi_{229}(120,\cdot)\) \(\chi_{229}(123,\cdot)\) \(\chi_{229}(128,\cdot)\) \(\chi_{229}(136,\cdot)\) \(\chi_{229}(141,\cdot)\) \(\chi_{229}(143,\cdot)\) \(\chi_{229}(145,\cdot)\) \(\chi_{229}(175,\cdot)\) \(\chi_{229}(177,\cdot)\) \(\chi_{229}(195,\cdot)\) \(\chi_{229}(197,\cdot)\) ...

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: $\Q(\zeta_{76})$
Fixed field: Number field defined by a degree 76 polynomial

Values on generators

\(6\) → \(e\left(\frac{13}{76}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 229 }(123, a) \) \(-1\)\(1\)\(e\left(\frac{45}{76}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{13}{76}\right)\)\(e\left(\frac{23}{76}\right)\)\(e\left(\frac{59}{76}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{27}{76}\right)\)\(e\left(\frac{27}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 229 }(123,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(123,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(123,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(123,·)) \;\) at \(\; a,b = \) e.g. 1,2