Properties

Label 462.47
Modulus $462$
Conductor $231$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(462, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,25,24]))
 
pari: [g,chi] = znchar(Mod(47,462))
 

Basic properties

Modulus: \(462\)
Conductor: \(231\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{231}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 462.bf

\(\chi_{462}(5,\cdot)\) \(\chi_{462}(47,\cdot)\) \(\chi_{462}(59,\cdot)\) \(\chi_{462}(185,\cdot)\) \(\chi_{462}(257,\cdot)\) \(\chi_{462}(269,\cdot)\) \(\chi_{462}(311,\cdot)\) \(\chi_{462}(383,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.189537114621750490998034780134822461022325002110684109.1

Values on generators

\((155,199,211)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)

Values

\(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 462 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 462 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 462 }(47,·),\chi_{ 462 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 462 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2