Properties

Label 299.64
Modulus $299$
Conductor $299$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(299, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,12]))
 
pari: [g,chi] = znchar(Mod(64,299))
 

Basic properties

Modulus: \(299\)
Conductor: \(299\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 299.p

\(\chi_{299}(12,\cdot)\) \(\chi_{299}(25,\cdot)\) \(\chi_{299}(64,\cdot)\) \(\chi_{299}(77,\cdot)\) \(\chi_{299}(142,\cdot)\) \(\chi_{299}(220,\cdot)\) \(\chi_{299}(233,\cdot)\) \(\chi_{299}(246,\cdot)\) \(\chi_{299}(259,\cdot)\) \(\chi_{299}(285,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((93,235)\) → \((-1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 299 }(64, a) \) \(1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 299 }(64,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 299 }(64,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 299 }(64,·),\chi_{ 299 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 299 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2