Properties

Label 283.34
Modulus $283$
Conductor $283$
Order $141$
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([49]))
 
pari: [g,chi] = znchar(Mod(34,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(141\)
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.g

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

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

\(3\) → \(e\left(\frac{49}{141}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{12}{47}\right)\)\(e\left(\frac{49}{141}\right)\)\(e\left(\frac{24}{47}\right)\)\(e\left(\frac{137}{141}\right)\)\(e\left(\frac{85}{141}\right)\)\(e\left(\frac{86}{141}\right)\)\(e\left(\frac{36}{47}\right)\)\(e\left(\frac{98}{141}\right)\)\(e\left(\frac{32}{141}\right)\)\(e\left(\frac{116}{141}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(34,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(34,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(34,r) e\left(\frac{2r}{283}\right) = 10.466497825+-13.1701337609i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(34,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(34,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(34,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(34,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(34,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = -3.9440212512+-4.0780879829i \)