Properties

Label 324.7
Modulus $324$
Conductor $324$
Order $54$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(324, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,16]))
 
pari: [g,chi] = znchar(Mod(7,324))
 

Basic properties

Modulus: \(324\)
Conductor: \(324\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 324.n

\(\chi_{324}(7,\cdot)\) \(\chi_{324}(31,\cdot)\) \(\chi_{324}(43,\cdot)\) \(\chi_{324}(67,\cdot)\) \(\chi_{324}(79,\cdot)\) \(\chi_{324}(103,\cdot)\) \(\chi_{324}(115,\cdot)\) \(\chi_{324}(139,\cdot)\) \(\chi_{324}(151,\cdot)\) \(\chi_{324}(175,\cdot)\) \(\chi_{324}(187,\cdot)\) \(\chi_{324}(211,\cdot)\) \(\chi_{324}(223,\cdot)\) \(\chi_{324}(247,\cdot)\) \(\chi_{324}(259,\cdot)\) \(\chi_{324}(283,\cdot)\) \(\chi_{324}(295,\cdot)\) \(\chi_{324}(319,\cdot)\)

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

\((163,245)\) → \((-1,e\left(\frac{8}{27}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{23}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 324 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 324 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 324 }(7,·),\chi_{ 324 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 324 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2