Properties

Conductor 71
Order 35
Real No
Primitive Yes
Parity Even
Orbit Label 71.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(71)
sage: chi = H[64]
pari: [g,chi] = znchar(Mod(64,71))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 71
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 35
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 = 71.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 ]

\(\chi_{71}(2,\cdot)\) \(\chi_{71}(3,\cdot)\) \(\chi_{71}(4,\cdot)\) \(\chi_{71}(6,\cdot)\) \(\chi_{71}(8,\cdot)\) \(\chi_{71}(9,\cdot)\) \(\chi_{71}(10,\cdot)\) \(\chi_{71}(12,\cdot)\) \(\chi_{71}(15,\cdot)\) \(\chi_{71}(16,\cdot)\) \(\chi_{71}(18,\cdot)\) \(\chi_{71}(19,\cdot)\) \(\chi_{71}(24,\cdot)\) \(\chi_{71}(27,\cdot)\) \(\chi_{71}(29,\cdot)\) \(\chi_{71}(36,\cdot)\) \(\chi_{71}(38,\cdot)\) \(\chi_{71}(40,\cdot)\) \(\chi_{71}(43,\cdot)\) \(\chi_{71}(49,\cdot)\) \(\chi_{71}(50,\cdot)\) \(\chi_{71}(58,\cdot)\) \(\chi_{71}(60,\cdot)\) \(\chi_{71}(64,\cdot)\)

Values on generators

\(7\) → \(e\left(\frac{18}{35}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{33}{35}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{35})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 71 }(64,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{71}(64,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(64,r) e\left(\frac{2r}{71}\right) = -0.5012586013+8.41122701i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 71 }(64,·),\chi_{ 71 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{71}(64,\cdot),\chi_{71}(1,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(64,r) \chi_{71}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 71 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{71}(64,·)) = \sum_{r \in \Z/71\Z} \chi_{71}(64,r) e\left(\frac{1 r + 2 r^{-1}}{71}\right) = -9.7663665833+-2.6953459147i \)