Properties

Conductor 229
Order 114
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 229.k

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[5]
 
pari: [g,chi] = znchar(Mod(5,229))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 229
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 114
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 = 229.k
Orbit index = 11

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 ]
 

\(\chi_{229}(5,\cdot)\) \(\chi_{229}(12,\cdot)\) \(\chi_{229}(33,\cdot)\) \(\chi_{229}(36,\cdot)\) \(\chi_{229}(45,\cdot)\) \(\chi_{229}(46,\cdot)\) \(\chi_{229}(49,\cdot)\) \(\chi_{229}(56,\cdot)\) \(\chi_{229}(58,\cdot)\) \(\chi_{229}(62,\cdot)\) \(\chi_{229}(70,\cdot)\) \(\chi_{229}(71,\cdot)\) \(\chi_{229}(76,\cdot)\) \(\chi_{229}(78,\cdot)\) \(\chi_{229}(80,\cdot)\) \(\chi_{229}(85,\cdot)\) \(\chi_{229}(97,\cdot)\) \(\chi_{229}(99,\cdot)\) \(\chi_{229}(100,\cdot)\) \(\chi_{229}(103,\cdot)\) \(\chi_{229}(118,\cdot)\) \(\chi_{229}(138,\cdot)\) \(\chi_{229}(146,\cdot)\) \(\chi_{229}(147,\cdot)\) \(\chi_{229}(148,\cdot)\) \(\chi_{229}(154,\cdot)\) \(\chi_{229}(174,\cdot)\) \(\chi_{229}(178,\cdot)\) \(\chi_{229}(181,\cdot)\) \(\chi_{229}(192,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{49}{114}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{113}{114}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{46}{57}\right)\)\(e\left(\frac{17}{114}\right)\)\(e\left(\frac{12}{19}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{57})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{229}(5,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(5,r) e\left(\frac{2r}{229}\right) = 8.9834245868+12.1777700214i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 229 }(5,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{229}(5,\cdot),\chi_{229}(1,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(5,r) \chi_{229}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{229}(5,·)) = \sum_{r \in \Z/229\Z} \chi_{229}(5,r) e\left(\frac{1 r + 2 r^{-1}}{229}\right) = -9.7781906451+-0.8102439773i \)