Properties

Label 966.79
Modulus $966$
Conductor $161$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,22,9]))
 
pari: [g,chi] = znchar(Mod(79,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.z

\(\chi_{966}(37,\cdot)\) \(\chi_{966}(67,\cdot)\) \(\chi_{966}(79,\cdot)\) \(\chi_{966}(109,\cdot)\) \(\chi_{966}(205,\cdot)\) \(\chi_{966}(235,\cdot)\) \(\chi_{966}(247,\cdot)\) \(\chi_{966}(319,\cdot)\) \(\chi_{966}(373,\cdot)\) \(\chi_{966}(457,\cdot)\) \(\chi_{966}(571,\cdot)\) \(\chi_{966}(613,\cdot)\) \(\chi_{966}(655,\cdot)\) \(\chi_{966}(697,\cdot)\) \(\chi_{966}(709,\cdot)\) \(\chi_{966}(751,\cdot)\) \(\chi_{966}(793,\cdot)\) \(\chi_{966}(835,\cdot)\) \(\chi_{966}(865,\cdot)\) \(\chi_{966}(907,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((323,829,925)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 966 }(79, a) \) \(-1\)\(1\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 966 }(79,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(79,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(79,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(79,·)) \;\) at \(\; a,b = \) e.g. 1,2