Properties

Conductor 229
Order 57
Real No
Primitive Yes
Parity Even
Orbit Label 229.i

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[14]
pari: [g,chi] = znchar(Mod(14,229))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 229
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 57
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 = 229.i
Orbit index = 9

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 ]

\(\chi_{229}(3,\cdot)\) \(\chi_{229}(9,\cdot)\) \(\chi_{229}(14,\cdot)\) \(\chi_{229}(19,\cdot)\) \(\chi_{229}(20,\cdot)\) \(\chi_{229}(25,\cdot)\) \(\chi_{229}(37,\cdot)\) \(\chi_{229}(48,\cdot)\) \(\chi_{229}(51,\cdot)\) \(\chi_{229}(55,\cdot)\) \(\chi_{229}(75,\cdot)\) \(\chi_{229}(81,\cdot)\) \(\chi_{229}(82,\cdot)\) \(\chi_{229}(83,\cdot)\) \(\chi_{229}(91,\cdot)\) \(\chi_{229}(111,\cdot)\) \(\chi_{229}(126,\cdot)\) \(\chi_{229}(129,\cdot)\) \(\chi_{229}(130,\cdot)\) \(\chi_{229}(132,\cdot)\) \(\chi_{229}(144,\cdot)\) \(\chi_{229}(149,\cdot)\) \(\chi_{229}(151,\cdot)\) \(\chi_{229}(153,\cdot)\) \(\chi_{229}(158,\cdot)\) \(\chi_{229}(159,\cdot)\) \(\chi_{229}(167,\cdot)\) \(\chi_{229}(171,\cdot)\) \(\chi_{229}(173,\cdot)\) \(\chi_{229}(180,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{32}{57}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{44}{57}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{1}{57}\right)\)\(e\left(\frac{32}{57}\right)\)\(e\left(\frac{4}{57}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{46}{57}\right)\)\(e\left(\frac{18}{19}\right)\)
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)
\( \tau_{ a }( \chi_{ 229 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{229}(14,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(14,r) e\left(\frac{2r}{229}\right) = 6.7269657535+-13.555365423i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 229 }(14,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{229}(14,\cdot),\chi_{229}(1,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(14,r) \chi_{229}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 229 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{229}(14,·)) = \sum_{r \in \Z/229\Z} \chi_{229}(14,r) e\left(\frac{1 r + 2 r^{-1}}{229}\right) = -2.0132903346+1.5670067664i \)