Properties

Label 283.68
Modulus $283$
Conductor $283$
Order $282$
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([257]))
 
pari: [g,chi] = znchar(Mod(68,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(282\)
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.h

\(\chi_{283}(3,\cdot)\) \(\chi_{283}(5,\cdot)\) \(\chi_{283}(12,\cdot)\) \(\chi_{283}(14,\cdot)\) \(\chi_{283}(17,\cdot)\) \(\chi_{283}(18,\cdot)\) \(\chi_{283}(20,\cdot)\) \(\chi_{283}(22,\cdot)\) \(\chi_{283}(26,\cdot)\) \(\chi_{283}(31,\cdot)\) \(\chi_{283}(35,\cdot)\) \(\chi_{283}(37,\cdot)\) \(\chi_{283}(46,\cdot)\) \(\chi_{283}(47,\cdot)\) \(\chi_{283}(48,\cdot)\) \(\chi_{283}(50,\cdot)\) \(\chi_{283}(55,\cdot)\) \(\chi_{283}(56,\cdot)\) \(\chi_{283}(65,\cdot)\) \(\chi_{283}(68,\cdot)\) \(\chi_{283}(69,\cdot)\) \(\chi_{283}(72,\cdot)\) \(\chi_{283}(75,\cdot)\) \(\chi_{283}(80,\cdot)\) \(\chi_{283}(82,\cdot)\) \(\chi_{283}(87,\cdot)\) \(\chi_{283}(88,\cdot)\) \(\chi_{283}(98,\cdot)\) \(\chi_{283}(104,\cdot)\) \(\chi_{283}(107,\cdot)\) ...

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

\(3\) → \(e\left(\frac{257}{282}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{85}{94}\right)\)\(e\left(\frac{257}{282}\right)\)\(e\left(\frac{38}{47}\right)\)\(e\left(\frac{97}{282}\right)\)\(e\left(\frac{115}{141}\right)\)\(e\left(\frac{50}{141}\right)\)\(e\left(\frac{67}{94}\right)\)\(e\left(\frac{116}{141}\right)\)\(e\left(\frac{35}{141}\right)\)\(e\left(\frac{74}{141}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{141})$
Fixed field: Number field defined by a degree 282 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(68,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(68,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(68,r) e\left(\frac{2r}{283}\right) = -3.4966980401+-16.4551846789i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(68,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(68,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(68,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(68,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(68,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 4.3518448191+14.0290260903i \)