Properties

Label 324.11
Modulus $324$
Conductor $324$
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(324, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,13]))
 
pari: [g,chi] = znchar(Mod(11,324))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 324.p

\(\chi_{324}(11,\cdot)\) \(\chi_{324}(23,\cdot)\) \(\chi_{324}(47,\cdot)\) \(\chi_{324}(59,\cdot)\) \(\chi_{324}(83,\cdot)\) \(\chi_{324}(95,\cdot)\) \(\chi_{324}(119,\cdot)\) \(\chi_{324}(131,\cdot)\) \(\chi_{324}(155,\cdot)\) \(\chi_{324}(167,\cdot)\) \(\chi_{324}(191,\cdot)\) \(\chi_{324}(203,\cdot)\) \(\chi_{324}(227,\cdot)\) \(\chi_{324}(239,\cdot)\) \(\chi_{324}(263,\cdot)\) \(\chi_{324}(275,\cdot)\) \(\chi_{324}(299,\cdot)\) \(\chi_{324}(311,\cdot)\)

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

\((163,245)\) → \((-1,e\left(\frac{13}{54}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{17}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 324 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 324 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 324 }(11,·),\chi_{ 324 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 324 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2