Properties

Label 966.635
Modulus $966$
Conductor $483$
Order $66$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,55,63]))
 
pari: [g,chi] = znchar(Mod(635,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{483}(152,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.bc

\(\chi_{966}(5,\cdot)\) \(\chi_{966}(17,\cdot)\) \(\chi_{966}(89,\cdot)\) \(\chi_{966}(143,\cdot)\) \(\chi_{966}(227,\cdot)\) \(\chi_{966}(341,\cdot)\) \(\chi_{966}(383,\cdot)\) \(\chi_{966}(425,\cdot)\) \(\chi_{966}(467,\cdot)\) \(\chi_{966}(479,\cdot)\) \(\chi_{966}(521,\cdot)\) \(\chi_{966}(563,\cdot)\) \(\chi_{966}(605,\cdot)\) \(\chi_{966}(635,\cdot)\) \(\chi_{966}(677,\cdot)\) \(\chi_{966}(773,\cdot)\) \(\chi_{966}(803,\cdot)\) \(\chi_{966}(815,\cdot)\) \(\chi_{966}(845,\cdot)\) \(\chi_{966}(941,\cdot)\)

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

\((323,829,925)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 966 }(635, a) \) \(-1\)\(1\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 966 }(635,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(635,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(635,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(635,·)) \;\) at \(\; a,b = \) e.g. 1,2