Properties

Conductor 137
Order 68
Real No
Primitive Yes
Parity Even
Orbit Label 137.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(137)
sage: chi = H[105]
pari: [g,chi] = znchar(Mod(105,137))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 137
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 68
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 = 137.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 ]

\(\chi_{137}(2,\cdot)\) \(\chi_{137}(7,\cdot)\) \(\chi_{137}(8,\cdot)\) \(\chi_{137}(9,\cdot)\) \(\chi_{137}(11,\cdot)\) \(\chi_{137}(17,\cdot)\) \(\chi_{137}(19,\cdot)\) \(\chi_{137}(25,\cdot)\) \(\chi_{137}(28,\cdot)\) \(\chi_{137}(30,\cdot)\) \(\chi_{137}(32,\cdot)\) \(\chi_{137}(36,\cdot)\) \(\chi_{137}(39,\cdot)\) \(\chi_{137}(44,\cdot)\) \(\chi_{137}(61,\cdot)\) \(\chi_{137}(68,\cdot)\) \(\chi_{137}(69,\cdot)\) \(\chi_{137}(76,\cdot)\) \(\chi_{137}(93,\cdot)\) \(\chi_{137}(98,\cdot)\) \(\chi_{137}(101,\cdot)\) \(\chi_{137}(105,\cdot)\) \(\chi_{137}(107,\cdot)\) \(\chi_{137}(109,\cdot)\) \(\chi_{137}(112,\cdot)\) \(\chi_{137}(118,\cdot)\) \(\chi_{137}(120,\cdot)\) \(\chi_{137}(126,\cdot)\) \(\chi_{137}(128,\cdot)\) \(\chi_{137}(129,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{59}{68}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{59}{68}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{5}{68}\right)\)\(e\left(\frac{37}{68}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{25}{34}\right)\)\(-i\)\(e\left(\frac{29}{34}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{68})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 137 }(105,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{137}(105,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(105,r) e\left(\frac{2r}{137}\right) = 11.1204612145+3.6517588058i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 137 }(105,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{137}(105,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(105,r) \chi_{137}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 137 }(105,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{137}(105,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(105,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = 1.7085857334+-2.759460707i \)