Properties

Label 666.611
Modulus $666$
Conductor $111$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,35]))
 
pari: [g,chi] = znchar(Mod(611,666))
 

Basic properties

Modulus: \(666\)
Conductor: \(111\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{111}(56,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 666.bs

\(\chi_{666}(17,\cdot)\) \(\chi_{666}(35,\cdot)\) \(\chi_{666}(89,\cdot)\) \(\chi_{666}(143,\cdot)\) \(\chi_{666}(161,\cdot)\) \(\chi_{666}(431,\cdot)\) \(\chi_{666}(449,\cdot)\) \(\chi_{666}(503,\cdot)\) \(\chi_{666}(557,\cdot)\) \(\chi_{666}(575,\cdot)\) \(\chi_{666}(611,\cdot)\) \(\chi_{666}(647,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{111})^+\)

Values on generators

\((371,631)\) → \((-1,e\left(\frac{35}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 666 }(611, a) \) \(1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 666 }(611,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 666 }(611,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 666 }(611,·),\chi_{ 666 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 666 }(611,·)) \;\) at \(\; a,b = \) e.g. 1,2