Properties

Label 285.62
Modulus $285$
Conductor $285$
Order $36$
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(285, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,9,32]))
 
pari: [g,chi] = znchar(Mod(62,285))
 

Basic properties

Modulus: \(285\)
Conductor: \(285\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 285.bi

\(\chi_{285}(17,\cdot)\) \(\chi_{285}(23,\cdot)\) \(\chi_{285}(47,\cdot)\) \(\chi_{285}(62,\cdot)\) \(\chi_{285}(92,\cdot)\) \(\chi_{285}(137,\cdot)\) \(\chi_{285}(158,\cdot)\) \(\chi_{285}(188,\cdot)\) \(\chi_{285}(218,\cdot)\) \(\chi_{285}(233,\cdot)\) \(\chi_{285}(263,\cdot)\) \(\chi_{285}(272,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.240152953708250935530977810544721792914847414233751595020294189453125.1

Values on generators

\((191,172,211)\) → \((-1,i,e\left(\frac{8}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\(1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{29}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 285 }(62,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 285 }(62,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 285 }(62,·),\chi_{ 285 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 285 }(62,·)) \;\) at \(\; a,b = \) e.g. 1,2