Properties

Label 536.21
Modulus $536$
Conductor $536$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,62]))
 
pari: [g,chi] = znchar(Mod(21,536))
 

Basic properties

Modulus: \(536\)
Conductor: \(536\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 536.bf

\(\chi_{536}(21,\cdot)\) \(\chi_{536}(77,\cdot)\) \(\chi_{536}(93,\cdot)\) \(\chi_{536}(157,\cdot)\) \(\chi_{536}(173,\cdot)\) \(\chi_{536}(181,\cdot)\) \(\chi_{536}(189,\cdot)\) \(\chi_{536}(205,\cdot)\) \(\chi_{536}(237,\cdot)\) \(\chi_{536}(261,\cdot)\) \(\chi_{536}(285,\cdot)\) \(\chi_{536}(301,\cdot)\) \(\chi_{536}(317,\cdot)\) \(\chi_{536}(333,\cdot)\) \(\chi_{536}(341,\cdot)\) \(\chi_{536}(389,\cdot)\) \(\chi_{536}(421,\cdot)\) \(\chi_{536}(437,\cdot)\) \(\chi_{536}(485,\cdot)\) \(\chi_{536}(525,\cdot)\)

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

\((135,269,337)\) → \((1,-1,e\left(\frac{31}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 536 }(21, a) \) \(1\)\(1\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{49}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 536 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 536 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 536 }(21,·),\chi_{ 536 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 536 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2