Properties

Label 648.11
Modulus $648$
Conductor $648$
Order $54$
Real no
Primitive yes
Minimal yes
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([27,27,13]))
 
pari: [g,chi] = znchar(Mod(11,648))
 

Basic properties

Modulus: \(648\)
Conductor: \(648\)
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 648.bb

\(\chi_{648}(11,\cdot)\) \(\chi_{648}(59,\cdot)\) \(\chi_{648}(83,\cdot)\) \(\chi_{648}(131,\cdot)\) \(\chi_{648}(155,\cdot)\) \(\chi_{648}(203,\cdot)\) \(\chi_{648}(227,\cdot)\) \(\chi_{648}(275,\cdot)\) \(\chi_{648}(299,\cdot)\) \(\chi_{648}(347,\cdot)\) \(\chi_{648}(371,\cdot)\) \(\chi_{648}(419,\cdot)\) \(\chi_{648}(443,\cdot)\) \(\chi_{648}(491,\cdot)\) \(\chi_{648}(515,\cdot)\) \(\chi_{648}(563,\cdot)\) \(\chi_{648}(587,\cdot)\) \(\chi_{648}(635,\cdot)\)

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

\((487,325,569)\) → \((-1,-1,e\left(\frac{13}{54}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{17}{54}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 648 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{648}(11,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(11,r) e\left(\frac{r}{324}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(11,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{648}(11,\cdot),\chi_{648}(1,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(11,r) \chi_{648}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{648}(11,·)) = \sum_{r \in \Z/648\Z} \chi_{648}(11,r) e\left(\frac{1 r + 2 r^{-1}}{648}\right) = -0.0 \)