Properties

Label 966.533
Modulus $966$
Conductor $69$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,4]))
 
pari: [g,chi] = znchar(Mod(533,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(50,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.w

\(\chi_{966}(29,\cdot)\) \(\chi_{966}(71,\cdot)\) \(\chi_{966}(197,\cdot)\) \(\chi_{966}(239,\cdot)\) \(\chi_{966}(407,\cdot)\) \(\chi_{966}(449,\cdot)\) \(\chi_{966}(491,\cdot)\) \(\chi_{966}(533,\cdot)\) \(\chi_{966}(785,\cdot)\) \(\chi_{966}(869,\cdot)\)

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: 22.0.304011857053427966889939263171547.1

Values on generators

\((323,829,925)\) → \((-1,1,e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 966 }(533, a) \) \(-1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 966 }(533,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(533,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(533,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(533,·)) \;\) at \(\; a,b = \) e.g. 1,2