Properties

Conductor 85
Order 16
Real No
Primitive Yes
Parity Even
Orbit Label 85.r

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(85)
sage: chi = H[28]
pari: [g,chi] = znchar(Mod(28,85))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 85
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 16
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 = 85.r
Orbit index = 18

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 ]

\(\chi_{85}(12,\cdot)\) \(\chi_{85}(22,\cdot)\) \(\chi_{85}(23,\cdot)\) \(\chi_{85}(28,\cdot)\) \(\chi_{85}(37,\cdot)\) \(\chi_{85}(58,\cdot)\) \(\chi_{85}(78,\cdot)\) \(\chi_{85}(82,\cdot)\)

Values on generators

\((52,71)\) → \((-i,e\left(\frac{7}{16}\right))\)

Values

-112346789111213
\(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{16})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 85 }(28,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{85}(28,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(28,r) e\left(\frac{2r}{85}\right) = -0.1951474635+-9.2174789106i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 85 }(28,·),\chi_{ 85 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{85}(28,\cdot),\chi_{85}(1,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(28,r) \chi_{85}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 85 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{85}(28,·)) = \sum_{r \in \Z/85\Z} \chi_{85}(28,r) e\left(\frac{1 r + 2 r^{-1}}{85}\right) = 7.6840461969+-3.1828361487i \)