Properties

Conductor 569
Order 284
Real No
Primitive Yes
Parity Even
Orbit Label 569.g

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(569)
 
sage: chi = H[68]
 
pari: [g,chi] = znchar(Mod(68,569))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 569
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 284
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 569.g
Orbit index = 7

Galois orbit

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

\(\chi_{569}(2,\cdot)\) \(\chi_{569}(7,\cdot)\) \(\chi_{569}(8,\cdot)\) \(\chi_{569}(9,\cdot)\) \(\chi_{569}(10,\cdot)\) \(\chi_{569}(13,\cdot)\) \(\chi_{569}(17,\cdot)\) \(\chi_{569}(28,\cdot)\) \(\chi_{569}(32,\cdot)\) \(\chi_{569}(35,\cdot)\) \(\chi_{569}(36,\cdot)\) \(\chi_{569}(40,\cdot)\) \(\chi_{569}(45,\cdot)\) \(\chi_{569}(50,\cdot)\) \(\chi_{569}(52,\cdot)\) \(\chi_{569}(57,\cdot)\) \(\chi_{569}(61,\cdot)\) \(\chi_{569}(65,\cdot)\) \(\chi_{569}(66,\cdot)\) \(\chi_{569}(67,\cdot)\) \(\chi_{569}(68,\cdot)\) \(\chi_{569}(71,\cdot)\) \(\chi_{569}(79,\cdot)\) \(\chi_{569}(82,\cdot)\) \(\chi_{569}(85,\cdot)\) \(\chi_{569}(98,\cdot)\) \(\chi_{569}(112,\cdot)\) \(\chi_{569}(121,\cdot)\) \(\chi_{569}(126,\cdot)\) \(\chi_{569}(127,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{241}{284}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{111}{142}\right)\)\(e\left(\frac{241}{284}\right)\)\(e\left(\frac{40}{71}\right)\)\(e\left(\frac{13}{71}\right)\)\(e\left(\frac{179}{284}\right)\)\(e\left(\frac{55}{142}\right)\)\(e\left(\frac{49}{142}\right)\)\(e\left(\frac{99}{142}\right)\)\(e\left(\frac{137}{142}\right)\)\(e\left(\frac{11}{284}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{284})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 569 }(68,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{569}(68,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(68,r) e\left(\frac{2r}{569}\right) = 7.8857687758+22.512544299i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 569 }(68,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{569}(68,\cdot),\chi_{569}(1,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(68,r) \chi_{569}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 569 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{569}(68,·)) = \sum_{r \in \Z/569\Z} \chi_{569}(68,r) e\left(\frac{1 r + 2 r^{-1}}{569}\right) = -16.9232636837+13.8494468365i \)