Properties

Conductor 37
Order 9
Real No
Primitive No
Parity Even
Orbit Label 74.f

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(74)
sage: chi = H[71]
pari: [g,chi] = znchar(Mod(71,74))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 37
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 9
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 = 74.f
Orbit index = 6

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 ]

\(\chi_{74}(7,\cdot)\) \(\chi_{74}(9,\cdot)\) \(\chi_{74}(33,\cdot)\) \(\chi_{74}(49,\cdot)\) \(\chi_{74}(53,\cdot)\) \(\chi_{74}(71,\cdot)\)

Inducing primitive character

\(\chi_{37}(34,\cdot)\)

Values on generators

\(39\) → \(e\left(\frac{2}{9}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 74 }(71,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{74}(71,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(71,r) e\left(\frac{r}{37}\right) = 4.391540734+4.2088442573i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 74 }(71,·),\chi_{ 74 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{74}(71,\cdot),\chi_{74}(1,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(71,r) \chi_{74}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 74 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{74}(71,·)) = \sum_{r \in \Z/74\Z} \chi_{74}(71,r) e\left(\frac{1 r + 2 r^{-1}}{74}\right) = -6.6896798048+-5.6133078569i \)