Properties

Conductor 269
Order 67
Real No
Primitive Yes
Parity Even
Orbit Label 269.d

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(269)
sage: chi = H[14]
pari: [g,chi] = znchar(Mod(14,269))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 269
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 67
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 = 269.d
Orbit index = 4

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 ]

\(\chi_{269}(5,\cdot)\) \(\chi_{269}(14,\cdot)\) \(\chi_{269}(16,\cdot)\) \(\chi_{269}(21,\cdot)\) \(\chi_{269}(23,\cdot)\) \(\chi_{269}(24,\cdot)\) \(\chi_{269}(25,\cdot)\) \(\chi_{269}(36,\cdot)\) \(\chi_{269}(37,\cdot)\) \(\chi_{269}(38,\cdot)\) \(\chi_{269}(41,\cdot)\) \(\chi_{269}(44,\cdot)\) \(\chi_{269}(47,\cdot)\) \(\chi_{269}(52,\cdot)\) \(\chi_{269}(53,\cdot)\) \(\chi_{269}(54,\cdot)\) \(\chi_{269}(57,\cdot)\) \(\chi_{269}(58,\cdot)\) \(\chi_{269}(61,\cdot)\) \(\chi_{269}(62,\cdot)\) \(\chi_{269}(66,\cdot)\) \(\chi_{269}(67,\cdot)\) \(\chi_{269}(70,\cdot)\) \(\chi_{269}(78,\cdot)\) \(\chi_{269}(80,\cdot)\) \(\chi_{269}(81,\cdot)\) \(\chi_{269}(87,\cdot)\) \(\chi_{269}(93,\cdot)\) \(\chi_{269}(99,\cdot)\) \(\chi_{269}(105,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{5}{67}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{5}{67}\right)\)\(e\left(\frac{9}{67}\right)\)\(e\left(\frac{10}{67}\right)\)\(e\left(\frac{35}{67}\right)\)\(e\left(\frac{14}{67}\right)\)\(e\left(\frac{28}{67}\right)\)\(e\left(\frac{15}{67}\right)\)\(e\left(\frac{18}{67}\right)\)\(e\left(\frac{40}{67}\right)\)\(e\left(\frac{11}{67}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{67})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 269 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{269}(14,\cdot)) = \sum_{r\in \Z/269\Z} \chi_{269}(14,r) e\left(\frac{2r}{269}\right) = -5.2421905777+15.540895661i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 269 }(14,·),\chi_{ 269 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{269}(14,\cdot),\chi_{269}(1,\cdot)) = \sum_{r\in \Z/269\Z} \chi_{269}(14,r) \chi_{269}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 269 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{269}(14,·)) = \sum_{r \in \Z/269\Z} \chi_{269}(14,r) e\left(\frac{1 r + 2 r^{-1}}{269}\right) = 4.2167414128+1.0071235932i \)