Properties

Label 448.347
Modulus $448$
Conductor $448$
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(448, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,27,32]))
 
pari: [g,chi] = znchar(Mod(347,448))
 

Basic properties

Modulus: \(448\)
Conductor: \(448\)
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 448.bl

\(\chi_{448}(11,\cdot)\) \(\chi_{448}(51,\cdot)\) \(\chi_{448}(67,\cdot)\) \(\chi_{448}(107,\cdot)\) \(\chi_{448}(123,\cdot)\) \(\chi_{448}(163,\cdot)\) \(\chi_{448}(179,\cdot)\) \(\chi_{448}(219,\cdot)\) \(\chi_{448}(235,\cdot)\) \(\chi_{448}(275,\cdot)\) \(\chi_{448}(291,\cdot)\) \(\chi_{448}(331,\cdot)\) \(\chi_{448}(347,\cdot)\) \(\chi_{448}(387,\cdot)\) \(\chi_{448}(403,\cdot)\) \(\chi_{448}(443,\cdot)\)

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

\((127,197,129)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 448 }(347, a) \) \(-1\)\(1\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{7}{16}\right)\)\(-i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{19}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 448 }(347,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 448 }(347,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 448 }(347,·),\chi_{ 448 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 448 }(347,·)) \;\) at \(\; a,b = \) e.g. 1,2