Properties

Conductor 43
Order 21
Real No
Primitive Yes
Parity Even
Orbit Label 43.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(43)
sage: chi = H[25]
pari: [g,chi] = znchar(Mod(25,43))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 43
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 21
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.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 ]

\(\chi_{43}(9,\cdot)\) \(\chi_{43}(10,\cdot)\) \(\chi_{43}(13,\cdot)\) \(\chi_{43}(14,\cdot)\) \(\chi_{43}(15,\cdot)\) \(\chi_{43}(17,\cdot)\) \(\chi_{43}(23,\cdot)\) \(\chi_{43}(24,\cdot)\) \(\chi_{43}(25,\cdot)\) \(\chi_{43}(31,\cdot)\) \(\chi_{43}(38,\cdot)\) \(\chi_{43}(40,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{4}{21}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{5}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 43 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{43}(25,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(25,r) e\left(\frac{2r}{43}\right) = 5.7064530571+3.2305407454i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 43 }(25,·),\chi_{ 43 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{43}(25,\cdot),\chi_{43}(1,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(25,r) \chi_{43}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 43 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{43}(25,·)) = \sum_{r \in \Z/43\Z} \chi_{43}(25,r) e\left(\frac{1 r + 2 r^{-1}}{43}\right) = -3.5013783387+-1.6861749387i \)