Properties

Label 567.104
Modulus $567$
Conductor $567$
Order $54$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,27]))
 
pari: [g,chi] = znchar(Mod(104,567))
 

Basic properties

Modulus: \(567\)
Conductor: \(567\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
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 567.bm

\(\chi_{567}(20,\cdot)\) \(\chi_{567}(41,\cdot)\) \(\chi_{567}(83,\cdot)\) \(\chi_{567}(104,\cdot)\) \(\chi_{567}(146,\cdot)\) \(\chi_{567}(167,\cdot)\) \(\chi_{567}(209,\cdot)\) \(\chi_{567}(230,\cdot)\) \(\chi_{567}(272,\cdot)\) \(\chi_{567}(293,\cdot)\) \(\chi_{567}(335,\cdot)\) \(\chi_{567}(356,\cdot)\) \(\chi_{567}(398,\cdot)\) \(\chi_{567}(419,\cdot)\) \(\chi_{567}(461,\cdot)\) \(\chi_{567}(482,\cdot)\) \(\chi_{567}(524,\cdot)\) \(\chi_{567}(545,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((407,325)\) → \((e\left(\frac{11}{54}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 567 }(104, a) \) \(1\)\(1\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 567 }(104,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 567 }(104,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 567 }(104,·),\chi_{ 567 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 567 }(104,·)) \;\) at \(\; a,b = \) e.g. 1,2