Properties

Label 224.157
Modulus $224$
Conductor $224$
Order $24$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,4]))
 
pari: [g,chi] = znchar(Mod(157,224))
 

Basic properties

Modulus: \(224\)
Conductor: \(224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
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 224.bc

\(\chi_{224}(5,\cdot)\) \(\chi_{224}(45,\cdot)\) \(\chi_{224}(61,\cdot)\) \(\chi_{224}(101,\cdot)\) \(\chi_{224}(117,\cdot)\) \(\chi_{224}(157,\cdot)\) \(\chi_{224}(173,\cdot)\) \(\chi_{224}(213,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.0.790224330201082600125157415256880139617697792.1

Values on generators

\((127,197,129)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 224 }(157, a) \) \(-1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 224 }(157,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 224 }(157,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 224 }(157,·),\chi_{ 224 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 224 }(157,·)) \;\) at \(\; a,b = \) e.g. 1,2