Properties

Label 283.42
Modulus $283$
Conductor $283$
Order $47$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(283)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([26]))
 
pari: [g,chi] = znchar(Mod(42,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(47\)
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 283.e

\(\chi_{283}(4,\cdot)\) \(\chi_{283}(15,\cdot)\) \(\chi_{283}(16,\cdot)\) \(\chi_{283}(29,\cdot)\) \(\chi_{283}(38,\cdot)\) \(\chi_{283}(42,\cdot)\) \(\chi_{283}(51,\cdot)\) \(\chi_{283}(54,\cdot)\) \(\chi_{283}(60,\cdot)\) \(\chi_{283}(61,\cdot)\) \(\chi_{283}(64,\cdot)\) \(\chi_{283}(66,\cdot)\) \(\chi_{283}(71,\cdot)\) \(\chi_{283}(78,\cdot)\) \(\chi_{283}(86,\cdot)\) \(\chi_{283}(106,\cdot)\) \(\chi_{283}(111,\cdot)\) \(\chi_{283}(116,\cdot)\) \(\chi_{283}(127,\cdot)\) \(\chi_{283}(134,\cdot)\) \(\chi_{283}(141,\cdot)\) \(\chi_{283}(151,\cdot)\) \(\chi_{283}(152,\cdot)\) \(\chi_{283}(155,\cdot)\) \(\chi_{283}(158,\cdot)\) \(\chi_{283}(161,\cdot)\) \(\chi_{283}(163,\cdot)\) \(\chi_{283}(168,\cdot)\) \(\chi_{283}(175,\cdot)\) \(\chi_{283}(181,\cdot)\) ...

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

Values on generators

\(3\) → \(e\left(\frac{26}{47}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{45}{47}\right)\)\(e\left(\frac{26}{47}\right)\)\(e\left(\frac{43}{47}\right)\)\(e\left(\frac{42}{47}\right)\)\(e\left(\frac{24}{47}\right)\)\(e\left(\frac{37}{47}\right)\)\(e\left(\frac{41}{47}\right)\)\(e\left(\frac{5}{47}\right)\)\(e\left(\frac{40}{47}\right)\)\(e\left(\frac{4}{47}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{47})$
Fixed field: Number field defined by a degree 47 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(42,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(42,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(42,r) e\left(\frac{2r}{283}\right) = 5.5867006207+-15.8678535465i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(42,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(42,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(42,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(42,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(42,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(42,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 24.7348792418+-3.3265176387i \)