Properties

Label 47.2
Modulus $47$
Conductor $47$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(47, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18]))
 
pari: [g,chi] = znchar(Mod(2,47))
 

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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{23})\)
Fixed field: \(\Q(\zeta_{47})^+\)

Values on generators

\(5\) → \(e\left(\frac{9}{23}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{17}{23}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 47 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{47}(2,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(2,r) e\left(\frac{2r}{47}\right) = 6.5113594751+-2.1452733592i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 47 }(2,·),\chi_{ 47 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{47}(2,\cdot),\chi_{47}(1,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(2,r) \chi_{47}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 47 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{47}(2,·)) = \sum_{r \in \Z/47\Z} \chi_{47}(2,r) e\left(\frac{1 r + 2 r^{-1}}{47}\right) = -6.1508410557+-0.845413644i \)