Properties

Label 384.197
Modulus $384$
Conductor $384$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,17,16]))
 
pari: [g,chi] = znchar(Mod(197,384))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 384.x

\(\chi_{384}(5,\cdot)\) \(\chi_{384}(29,\cdot)\) \(\chi_{384}(53,\cdot)\) \(\chi_{384}(77,\cdot)\) \(\chi_{384}(101,\cdot)\) \(\chi_{384}(125,\cdot)\) \(\chi_{384}(149,\cdot)\) \(\chi_{384}(173,\cdot)\) \(\chi_{384}(197,\cdot)\) \(\chi_{384}(221,\cdot)\) \(\chi_{384}(245,\cdot)\) \(\chi_{384}(269,\cdot)\) \(\chi_{384}(293,\cdot)\) \(\chi_{384}(317,\cdot)\) \(\chi_{384}(341,\cdot)\) \(\chi_{384}(365,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{32})\)
Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((127,133,257)\) → \((1,e\left(\frac{17}{32}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 384 }(197, a) \) \(-1\)\(1\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{27}{32}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 384 }(197,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(197,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(197,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(197,·)) \;\) at \(\; a,b = \) e.g. 1,2