Properties

Label 47.28
Modulus $47$
Conductor $47$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([22]))
 
pari: [g,chi] = znchar(Mod(28,47))
 

Basic properties

Modulus: \(47\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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 47.c

\(\chi_{47}(2,\cdot)\) \(\chi_{47}(3,\cdot)\) \(\chi_{47}(4,\cdot)\) \(\chi_{47}(6,\cdot)\) \(\chi_{47}(7,\cdot)\) \(\chi_{47}(8,\cdot)\) \(\chi_{47}(9,\cdot)\) \(\chi_{47}(12,\cdot)\) \(\chi_{47}(14,\cdot)\) \(\chi_{47}(16,\cdot)\) \(\chi_{47}(17,\cdot)\) \(\chi_{47}(18,\cdot)\) \(\chi_{47}(21,\cdot)\) \(\chi_{47}(24,\cdot)\) \(\chi_{47}(25,\cdot)\) \(\chi_{47}(27,\cdot)\) \(\chi_{47}(28,\cdot)\) \(\chi_{47}(32,\cdot)\) \(\chi_{47}(34,\cdot)\) \(\chi_{47}(36,\cdot)\) \(\chi_{47}(37,\cdot)\) \(\chi_{47}(42,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(5\) → \(e\left(\frac{11}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 47 }(28, a) \) \(1\)\(1\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{8}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 47 }(28,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 47 }(28,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 47 }(28,·),\chi_{ 47 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 47 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2