Properties

Label 324.5
Modulus $324$
Conductor $81$
Order $54$
Real no
Primitive no
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([0,23]))
 
pari: [g,chi] = znchar(Mod(5,324))
 

Basic properties

Modulus: \(324\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 324.o

\(\chi_{324}(5,\cdot)\) \(\chi_{324}(29,\cdot)\) \(\chi_{324}(41,\cdot)\) \(\chi_{324}(65,\cdot)\) \(\chi_{324}(77,\cdot)\) \(\chi_{324}(101,\cdot)\) \(\chi_{324}(113,\cdot)\) \(\chi_{324}(137,\cdot)\) \(\chi_{324}(149,\cdot)\) \(\chi_{324}(173,\cdot)\) \(\chi_{324}(185,\cdot)\) \(\chi_{324}(209,\cdot)\) \(\chi_{324}(221,\cdot)\) \(\chi_{324}(245,\cdot)\) \(\chi_{324}(257,\cdot)\) \(\chi_{324}(281,\cdot)\) \(\chi_{324}(293,\cdot)\) \(\chi_{324}(317,\cdot)\)

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

\((163,245)\) → \((1,e\left(\frac{23}{54}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{14}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 324 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 324 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 324 }(5,·),\chi_{ 324 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 324 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2