Properties

Modulus 283
Conductor 283
Order 141
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 283.g

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([127]))
 
pari: [g,chi] = znchar(Mod(13,283))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 283
Conductor = 283
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 141
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 283.g
Orbit index = 7

Galois orbit

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

\(\chi_{283}(6,\cdot)\) \(\chi_{283}(7,\cdot)\) \(\chi_{283}(9,\cdot)\) \(\chi_{283}(10,\cdot)\) \(\chi_{283}(11,\cdot)\) \(\chi_{283}(13,\cdot)\) \(\chi_{283}(23,\cdot)\) \(\chi_{283}(24,\cdot)\) \(\chi_{283}(25,\cdot)\) \(\chi_{283}(28,\cdot)\) \(\chi_{283}(34,\cdot)\) \(\chi_{283}(36,\cdot)\) \(\chi_{283}(40,\cdot)\) \(\chi_{283}(41,\cdot)\) \(\chi_{283}(49,\cdot)\) \(\chi_{283}(52,\cdot)\) \(\chi_{283}(57,\cdot)\) \(\chi_{283}(59,\cdot)\) \(\chi_{283}(62,\cdot)\) \(\chi_{283}(63,\cdot)\) \(\chi_{283}(70,\cdot)\) \(\chi_{283}(73,\cdot)\) \(\chi_{283}(74,\cdot)\) \(\chi_{283}(77,\cdot)\) \(\chi_{283}(81,\cdot)\) \(\chi_{283}(83,\cdot)\) \(\chi_{283}(85,\cdot)\) \(\chi_{283}(89,\cdot)\) \(\chi_{283}(90,\cdot)\) \(\chi_{283}(91,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{127}{141}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{10}{47}\right)\)\(e\left(\frac{127}{141}\right)\)\(e\left(\frac{20}{47}\right)\)\(e\left(\frac{122}{141}\right)\)\(e\left(\frac{16}{141}\right)\)\(e\left(\frac{56}{141}\right)\)\(e\left(\frac{30}{47}\right)\)\(e\left(\frac{113}{141}\right)\)\(e\left(\frac{11}{141}\right)\)\(e\left(\frac{128}{141}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{141})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(13,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(13,r) e\left(\frac{2r}{283}\right) = 5.8684257655+-15.7658358178i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(13,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(13,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(13,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(13,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(13,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 1.5967897737+1.2609722988i \)