Properties

Label 648.157
Modulus $648$
Conductor $648$
Order $54$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: 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([0,27,50]))
 
pari: [g,chi] = znchar(Mod(157,648))
 

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.bd

\(\chi_{648}(13,\cdot)\) \(\chi_{648}(61,\cdot)\) \(\chi_{648}(85,\cdot)\) \(\chi_{648}(133,\cdot)\) \(\chi_{648}(157,\cdot)\) \(\chi_{648}(205,\cdot)\) \(\chi_{648}(229,\cdot)\) \(\chi_{648}(277,\cdot)\) \(\chi_{648}(301,\cdot)\) \(\chi_{648}(349,\cdot)\) \(\chi_{648}(373,\cdot)\) \(\chi_{648}(421,\cdot)\) \(\chi_{648}(445,\cdot)\) \(\chi_{648}(493,\cdot)\) \(\chi_{648}(517,\cdot)\) \(\chi_{648}(565,\cdot)\) \(\chi_{648}(589,\cdot)\) \(\chi_{648}(637,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((487,325,569)\) → \((1,-1,e\left(\frac{25}{27}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{14}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 648 }(157,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(157,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(157,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(157,·)) \;\) at \(\; a,b = \) e.g. 1,2