Properties

Conductor 139
Order 69
Real No
Primitive Yes
Parity Even
Orbit Label 139.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(139)
sage: chi = H[37]
pari: [g,chi] = znchar(Mod(37,139))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 139
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 69
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 = 139.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 ]

\(\chi_{139}(4,\cdot)\) \(\chi_{139}(5,\cdot)\) \(\chi_{139}(7,\cdot)\) \(\chi_{139}(9,\cdot)\) \(\chi_{139}(11,\cdot)\) \(\chi_{139}(13,\cdot)\) \(\chi_{139}(16,\cdot)\) \(\chi_{139}(20,\cdot)\) \(\chi_{139}(24,\cdot)\) \(\chi_{139}(25,\cdot)\) \(\chi_{139}(28,\cdot)\) \(\chi_{139}(29,\cdot)\) \(\chi_{139}(30,\cdot)\) \(\chi_{139}(31,\cdot)\) \(\chi_{139}(35,\cdot)\) \(\chi_{139}(37,\cdot)\) \(\chi_{139}(38,\cdot)\) \(\chi_{139}(41,\cdot)\) \(\chi_{139}(46,\cdot)\) \(\chi_{139}(47,\cdot)\) \(\chi_{139}(49,\cdot)\) \(\chi_{139}(51,\cdot)\) \(\chi_{139}(54,\cdot)\) \(\chi_{139}(66,\cdot)\) \(\chi_{139}(67,\cdot)\) \(\chi_{139}(69,\cdot)\) \(\chi_{139}(71,\cdot)\) \(\chi_{139}(78,\cdot)\) \(\chi_{139}(81,\cdot)\) \(\chi_{139}(83,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{40}{69}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{40}{69}\right)\)\(e\left(\frac{53}{69}\right)\)\(e\left(\frac{11}{69}\right)\)\(e\left(\frac{59}{69}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{68}{69}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{37}{69}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{4}{69}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{69})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 139 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{139}(37,\cdot)) = \sum_{r\in \Z/139\Z} \chi_{139}(37,r) e\left(\frac{2r}{139}\right) = 6.7051991135+9.6974380559i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 139 }(37,·),\chi_{ 139 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{139}(37,\cdot),\chi_{139}(1,\cdot)) = \sum_{r\in \Z/139\Z} \chi_{139}(37,r) \chi_{139}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 139 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{139}(37,·)) = \sum_{r \in \Z/139\Z} \chi_{139}(37,r) e\left(\frac{1 r + 2 r^{-1}}{139}\right) = 0.7826586573+-3.0598187979i \)