Properties

Label 384.347
Modulus $384$
Conductor $384$
Order $32$
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(384, base_ring=CyclotomicField(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([16,25,16]))
 
pari: [g,chi] = znchar(Mod(347,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 384.w

\(\chi_{384}(11,\cdot)\) \(\chi_{384}(35,\cdot)\) \(\chi_{384}(59,\cdot)\) \(\chi_{384}(83,\cdot)\) \(\chi_{384}(107,\cdot)\) \(\chi_{384}(131,\cdot)\) \(\chi_{384}(155,\cdot)\) \(\chi_{384}(179,\cdot)\) \(\chi_{384}(203,\cdot)\) \(\chi_{384}(227,\cdot)\) \(\chi_{384}(251,\cdot)\) \(\chi_{384}(275,\cdot)\) \(\chi_{384}(299,\cdot)\) \(\chi_{384}(323,\cdot)\) \(\chi_{384}(347,\cdot)\) \(\chi_{384}(371,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.32.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((127,133,257)\) → \((-1,e\left(\frac{25}{32}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 384 }(347,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(347,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(347,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(347,·)) \;\) at \(\; a,b = \) e.g. 1,2