Properties

Label 648.23
Modulus $648$
Conductor $324$
Order $54$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(648, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,0,11]))
 
pari: [g,chi] = znchar(Mod(23,648))
 

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 \(\chi_{324}(23,\cdot)\)
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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((487,325,569)\) → \((-1,1,e\left(\frac{11}{54}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{31}{54}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{648}(23,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) e\left(\frac{r}{324}\right) = -17.3920535747+-31.5200963269i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(23,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{648}(23,\cdot),\chi_{648}(1,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) \chi_{648}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{648}(23,·)) = \sum_{r \in \Z/648\Z} \chi_{648}(23,r) e\left(\frac{1 r + 2 r^{-1}}{648}\right) = -0.0 \)