Properties

Conductor 47
Order 46
Real No
Primitive Yes
Parity Odd
Orbit Label 47.d

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(47)
sage: chi = H[31]
pari: [g,chi] = znchar(Mod(31,47))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 47
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 46
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 = Odd
Orbit label = 47.d
Orbit index = 4

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 ]

\(\chi_{47}(5,\cdot)\) \(\chi_{47}(10,\cdot)\) \(\chi_{47}(11,\cdot)\) \(\chi_{47}(13,\cdot)\) \(\chi_{47}(15,\cdot)\) \(\chi_{47}(19,\cdot)\) \(\chi_{47}(20,\cdot)\) \(\chi_{47}(22,\cdot)\) \(\chi_{47}(23,\cdot)\) \(\chi_{47}(26,\cdot)\) \(\chi_{47}(29,\cdot)\) \(\chi_{47}(30,\cdot)\) \(\chi_{47}(31,\cdot)\) \(\chi_{47}(33,\cdot)\) \(\chi_{47}(35,\cdot)\) \(\chi_{47}(38,\cdot)\) \(\chi_{47}(39,\cdot)\) \(\chi_{47}(40,\cdot)\) \(\chi_{47}(41,\cdot)\) \(\chi_{47}(43,\cdot)\) \(\chi_{47}(44,\cdot)\) \(\chi_{47}(45,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{3}{46}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{11}{46}\right)\)\(e\left(\frac{21}{46}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{23})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 47 }(31,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{47}(31,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(31,r) e\left(\frac{2r}{47}\right) = -6.7538206675+-1.1772452553i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 47 }(31,·),\chi_{ 47 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{47}(31,\cdot),\chi_{47}(1,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(31,r) \chi_{47}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 47 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{47}(31,·)) = \sum_{r \in \Z/47\Z} \chi_{47}(31,r) e\left(\frac{1 r + 2 r^{-1}}{47}\right) = 2.9132147818+-4.7905776131i \)