Properties

Label 388.31
Modulus $388$
Conductor $388$
Order $48$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(388, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,23]))
 
pari: [g,chi] = znchar(Mod(31,388))
 

Basic properties

Modulus: \(388\)
Conductor: \(388\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 388.v

\(\chi_{388}(3,\cdot)\) \(\chi_{388}(11,\cdot)\) \(\chi_{388}(31,\cdot)\) \(\chi_{388}(95,\cdot)\) \(\chi_{388}(99,\cdot)\) \(\chi_{388}(163,\cdot)\) \(\chi_{388}(183,\cdot)\) \(\chi_{388}(191,\cdot)\) \(\chi_{388}(219,\cdot)\) \(\chi_{388}(243,\cdot)\) \(\chi_{388}(247,\cdot)\) \(\chi_{388}(259,\cdot)\) \(\chi_{388}(323,\cdot)\) \(\chi_{388}(335,\cdot)\) \(\chi_{388}(339,\cdot)\) \(\chi_{388}(363,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((195,5)\) → \((-1,e\left(\frac{23}{48}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 388 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{19}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 388 }(31,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 388 }(31,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 388 }(31,·),\chi_{ 388 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 388 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2