Properties

Label 388.127
Modulus $388$
Conductor $388$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(388, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,3]))
 
pari: [g,chi] = znchar(Mod(127,388))
 

Basic properties

Modulus: \(388\)
Conductor: \(388\)
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 388.s

\(\chi_{388}(19,\cdot)\) \(\chi_{388}(51,\cdot)\) \(\chi_{388}(55,\cdot)\) \(\chi_{388}(63,\cdot)\) \(\chi_{388}(67,\cdot)\) \(\chi_{388}(127,\cdot)\) \(\chi_{388}(131,\cdot)\) \(\chi_{388}(139,\cdot)\) \(\chi_{388}(143,\cdot)\) \(\chi_{388}(175,\cdot)\) \(\chi_{388}(239,\cdot)\) \(\chi_{388}(263,\cdot)\) \(\chi_{388}(271,\cdot)\) \(\chi_{388}(311,\cdot)\) \(\chi_{388}(319,\cdot)\) \(\chi_{388}(343,\cdot)\)

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: 32.32.167064287751196233763024674646775194239336663364043996417475105771225088.1

Values on generators

\((195,5)\) → \((-1,e\left(\frac{3}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 388 }(127, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{15}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 388 }(127,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 388 }(127,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 388 }(127,·),\chi_{ 388 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 388 }(127,·)) \;\) at \(\; a,b = \) e.g. 1,2