Properties

Conductor 49
Order 7
Real No
Primitive Yes
Parity Even
Orbit Label 49.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(49)
sage: chi = H[43]
pari: [g,chi] = znchar(Mod(43,49))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 49
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 = 49.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 ]

\(\chi_{49}(8,\cdot)\) \(\chi_{49}(15,\cdot)\) \(\chi_{49}(22,\cdot)\) \(\chi_{49}(29,\cdot)\) \(\chi_{49}(36,\cdot)\) \(\chi_{49}(43,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{1}{7}\right)\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)
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)
\( \tau_{ a }( \chi_{ 49 }(43,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{49}(43,\cdot)) = \sum_{r\in \Z/49\Z} \chi_{49}(43,r) e\left(\frac{2r}{49}\right) = -0.6721611814+6.9676537906i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 49 }(43,·),\chi_{ 49 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{49}(43,\cdot),\chi_{49}(1,\cdot)) = \sum_{r\in \Z/49\Z} \chi_{49}(43,r) \chi_{49}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 49 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{49}(43,·)) = \sum_{r \in \Z/49\Z} \chi_{49}(43,r) e\left(\frac{1 r + 2 r^{-1}}{49}\right) = 2.4836006806+-3.114336748i \)