Properties

Conductor 43
Order 7
Real No
Primitive Yes
Parity Even
Orbit Label 43.e

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(43)
sage: chi = H[21]
pari: [g,chi] = znchar(Mod(21,43))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 43
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 7
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 = 43.e
Orbit index = 5

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 ]

\(\chi_{43}(4,\cdot)\) \(\chi_{43}(11,\cdot)\) \(\chi_{43}(16,\cdot)\) \(\chi_{43}(21,\cdot)\) \(\chi_{43}(35,\cdot)\) \(\chi_{43}(41,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{6}{7}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(1\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{7})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 43 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{43}(21,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(21,r) e\left(\frac{2r}{43}\right) = 6.3839752672+-1.4982856164i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 43 }(21,·),\chi_{ 43 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{43}(21,\cdot),\chi_{43}(1,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(21,r) \chi_{43}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 43 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{43}(21,·)) = \sum_{r \in \Z/43\Z} \chi_{43}(21,r) e\left(\frac{1 r + 2 r^{-1}}{43}\right) = 5.2885052421+2.546809896i \)