Properties

Label 667.28
Modulus $667$
Conductor $667$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,11]))
 
pari: [g,chi] = znchar(Mod(28,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 667.n

\(\chi_{667}(28,\cdot)\) \(\chi_{667}(57,\cdot)\) \(\chi_{667}(86,\cdot)\) \(\chi_{667}(260,\cdot)\) \(\chi_{667}(318,\cdot)\) \(\chi_{667}(405,\cdot)\) \(\chi_{667}(434,\cdot)\) \(\chi_{667}(521,\cdot)\) \(\chi_{667}(550,\cdot)\) \(\chi_{667}(608,\cdot)\)

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: 22.0.481573447532424402907134681295965659189012467.1

Values on generators

\((465,553)\) → \((e\left(\frac{1}{22}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 667 }(28, a) \) \(-1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 667 }(28,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(28,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(28,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2