Properties

Label 462.79
Modulus $462$
Conductor $77$
Order $30$
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(462, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,3]))
 
pari: [g,chi] = znchar(Mod(79,462))
 

Basic properties

Modulus: \(462\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 462.be

\(\chi_{462}(79,\cdot)\) \(\chi_{462}(151,\cdot)\) \(\chi_{462}(193,\cdot)\) \(\chi_{462}(205,\cdot)\) \(\chi_{462}(277,\cdot)\) \(\chi_{462}(403,\cdot)\) \(\chi_{462}(415,\cdot)\) \(\chi_{462}(457,\cdot)\)

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: 30.0.1046076147688308987260717152173116396995512371.1

Values on generators

\((155,199,211)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 462 }(79,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 462 }(79,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 462 }(79,·),\chi_{ 462 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 462 }(79,·)) \;\) at \(\; a,b = \) e.g. 1,2