Properties

Conductor 7
Order 3
Real No
Primitive No
Parity Even
Orbit Label 21.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(21)
sage: chi = H[4]
pari: [g,chi] = znchar(Mod(4,21))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 7
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 3
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 21.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 ]

\(\chi_{21}(4,\cdot)\) \(\chi_{21}(16,\cdot)\)

Inducing primitive character

\(\chi_{7}(4,\cdot)\)

Values on generators

\((8,10)\) → \((1,e\left(\frac{2}{3}\right))\)

Values

-112458101113161719
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 21 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{21}(4,\cdot)) = \sum_{r\in \Z/21\Z} \chi_{21}(4,r) e\left(\frac{2r}{21}\right) = 2.2029066037+-1.4653335782i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 21 }(4,·),\chi_{ 21 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{21}(4,\cdot),\chi_{21}(1,\cdot)) = \sum_{r\in \Z/21\Z} \chi_{21}(4,r) \chi_{21}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 21 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{21}(4,·)) = \sum_{r \in \Z/21\Z} \chi_{21}(4,r) e\left(\frac{1 r + 2 r^{-1}}{21}\right) = 1.6920214716+2.9306671564i \)