Properties

Label 648.455
Modulus $648$
Conductor $324$
Order $54$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,0,47]))
 
pari: [g,chi] = znchar(Mod(455,648))
 

Basic properties

Modulus: \(648\)
Conductor: \(324\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{324}(131,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 648.be

\(\chi_{648}(23,\cdot)\) \(\chi_{648}(47,\cdot)\) \(\chi_{648}(95,\cdot)\) \(\chi_{648}(119,\cdot)\) \(\chi_{648}(167,\cdot)\) \(\chi_{648}(191,\cdot)\) \(\chi_{648}(239,\cdot)\) \(\chi_{648}(263,\cdot)\) \(\chi_{648}(311,\cdot)\) \(\chi_{648}(335,\cdot)\) \(\chi_{648}(383,\cdot)\) \(\chi_{648}(407,\cdot)\) \(\chi_{648}(455,\cdot)\) \(\chi_{648}(479,\cdot)\) \(\chi_{648}(527,\cdot)\) \(\chi_{648}(551,\cdot)\) \(\chi_{648}(599,\cdot)\) \(\chi_{648}(623,\cdot)\)

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

\((487,325,569)\) → \((-1,1,e\left(\frac{47}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 648 }(455, a) \) \(1\)\(1\)\(e\left(\frac{1}{54}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{49}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 648 }(455,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(455,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(455,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(455,·)) \;\) at \(\; a,b = \) e.g. 1,2