Properties

Label 283.19
Modulus $283$
Conductor $283$
Order $94$
Real no
Primitive yes
Minimal yes
Parity odd

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([69]))
 
pari: [g,chi] = znchar(Mod(19,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(94\)
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 283.f

\(\chi_{283}(2,\cdot)\) \(\chi_{283}(8,\cdot)\) \(\chi_{283}(19,\cdot)\) \(\chi_{283}(21,\cdot)\) \(\chi_{283}(27,\cdot)\) \(\chi_{283}(30,\cdot)\) \(\chi_{283}(32,\cdot)\) \(\chi_{283}(33,\cdot)\) \(\chi_{283}(39,\cdot)\) \(\chi_{283}(43,\cdot)\) \(\chi_{283}(53,\cdot)\) \(\chi_{283}(58,\cdot)\) \(\chi_{283}(67,\cdot)\) \(\chi_{283}(76,\cdot)\) \(\chi_{283}(79,\cdot)\) \(\chi_{283}(84,\cdot)\) \(\chi_{283}(102,\cdot)\) \(\chi_{283}(108,\cdot)\) \(\chi_{283}(115,\cdot)\) \(\chi_{283}(120,\cdot)\) \(\chi_{283}(122,\cdot)\) \(\chi_{283}(125,\cdot)\) \(\chi_{283}(128,\cdot)\) \(\chi_{283}(131,\cdot)\) \(\chi_{283}(132,\cdot)\) \(\chi_{283}(142,\cdot)\) \(\chi_{283}(149,\cdot)\) \(\chi_{283}(156,\cdot)\) \(\chi_{283}(167,\cdot)\) \(\chi_{283}(172,\cdot)\) ...

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

\(3\) → \(e\left(\frac{69}{94}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{67}{94}\right)\)\(e\left(\frac{69}{94}\right)\)\(e\left(\frac{20}{47}\right)\)\(e\left(\frac{3}{94}\right)\)\(e\left(\frac{21}{47}\right)\)\(e\left(\frac{3}{47}\right)\)\(e\left(\frac{13}{94}\right)\)\(e\left(\frac{22}{47}\right)\)\(e\left(\frac{35}{47}\right)\)\(e\left(\frac{27}{47}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{47})$
Fixed field: Number field defined by a degree 94 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(19,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(19,r) e\left(\frac{2r}{283}\right) = -16.230745273+-4.4229976128i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(19,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(19,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(19,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(19,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(19,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 3.9853244226+3.1471792852i \)