Properties

Label 384.277
Modulus $384$
Conductor $128$
Order $32$
Real no
Primitive no
Minimal yes
Parity even

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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([0,13,0]))
 
pari: [g,chi] = znchar(Mod(277,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(128\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{128}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 384.v

\(\chi_{384}(13,\cdot)\) \(\chi_{384}(37,\cdot)\) \(\chi_{384}(61,\cdot)\) \(\chi_{384}(85,\cdot)\) \(\chi_{384}(109,\cdot)\) \(\chi_{384}(133,\cdot)\) \(\chi_{384}(157,\cdot)\) \(\chi_{384}(181,\cdot)\) \(\chi_{384}(205,\cdot)\) \(\chi_{384}(229,\cdot)\) \(\chi_{384}(253,\cdot)\) \(\chi_{384}(277,\cdot)\) \(\chi_{384}(301,\cdot)\) \(\chi_{384}(325,\cdot)\) \(\chi_{384}(349,\cdot)\) \(\chi_{384}(373,\cdot)\)

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: \(\Q(\zeta_{128})^+\)

Values on generators

\((127,133,257)\) → \((1,e\left(\frac{13}{32}\right),1)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 384 }(277, a) \) \(1\)\(1\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{31}{32}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 384 }(277,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(277,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(277,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(277,·)) \;\) at \(\; a,b = \) e.g. 1,2