Properties

Conductor 229
Order 228
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 229.l

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(229)
 
sage: chi = H[31]
 
pari: [g,chi] = znchar(Mod(31,229))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 229
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 228
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 = odd
Orbit label = 229.l
Orbit index = 12

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 ]
 

\(\chi_{229}(6,\cdot)\) \(\chi_{229}(7,\cdot)\) \(\chi_{229}(10,\cdot)\) \(\chi_{229}(23,\cdot)\) \(\chi_{229}(24,\cdot)\) \(\chi_{229}(28,\cdot)\) \(\chi_{229}(29,\cdot)\) \(\chi_{229}(31,\cdot)\) \(\chi_{229}(35,\cdot)\) \(\chi_{229}(38,\cdot)\) \(\chi_{229}(39,\cdot)\) \(\chi_{229}(40,\cdot)\) \(\chi_{229}(41,\cdot)\) \(\chi_{229}(47,\cdot)\) \(\chi_{229}(50,\cdot)\) \(\chi_{229}(59,\cdot)\) \(\chi_{229}(63,\cdot)\) \(\chi_{229}(65,\cdot)\) \(\chi_{229}(66,\cdot)\) \(\chi_{229}(67,\cdot)\) \(\chi_{229}(69,\cdot)\) \(\chi_{229}(72,\cdot)\) \(\chi_{229}(73,\cdot)\) \(\chi_{229}(74,\cdot)\) \(\chi_{229}(77,\cdot)\) \(\chi_{229}(79,\cdot)\) \(\chi_{229}(87,\cdot)\) \(\chi_{229}(90,\cdot)\) \(\chi_{229}(92,\cdot)\) \(\chi_{229}(96,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{29}{228}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{51}{76}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{13}{38}\right)\)\(e\left(\frac{53}{114}\right)\)\(e\left(\frac{29}{228}\right)\)\(e\left(\frac{139}{228}\right)\)\(e\left(\frac{1}{76}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{31}{228}\right)\)\(e\left(\frac{23}{38}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{228})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(31,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{229}(31,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(31,r) e\left(\frac{2r}{229}\right) = 3.6143206916+-14.6947843107i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 229 }(31,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{229}(31,\cdot),\chi_{229}(1,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(31,r) \chi_{229}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{229}(31,·)) = \sum_{r \in \Z/229\Z} \chi_{229}(31,r) e\left(\frac{1 r + 2 r^{-1}}{229}\right) = 17.8676969705+10.6468364348i \)