Properties

Conductor 87
Order 28
Real No
Primitive Yes
Parity Even
Orbit Label 87.k

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(87)
sage: chi = H[8]
pari: [g,chi] = znchar(Mod(8,87))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 87
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 28
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 = 87.k
Orbit index = 11

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 ]

\(\chi_{87}(2,\cdot)\) \(\chi_{87}(8,\cdot)\) \(\chi_{87}(11,\cdot)\) \(\chi_{87}(14,\cdot)\) \(\chi_{87}(26,\cdot)\) \(\chi_{87}(32,\cdot)\) \(\chi_{87}(44,\cdot)\) \(\chi_{87}(47,\cdot)\) \(\chi_{87}(50,\cdot)\) \(\chi_{87}(56,\cdot)\) \(\chi_{87}(68,\cdot)\) \(\chi_{87}(77,\cdot)\)

Values on generators

\((59,31)\) → \((-1,e\left(\frac{3}{28}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{3}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{28})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 87 }(8,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{87}(8,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(8,r) e\left(\frac{2r}{87}\right) = -2.3611785129+9.0235711351i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 87 }(8,·),\chi_{ 87 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{87}(8,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(8,r) \chi_{87}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 87 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{87}(8,·)) = \sum_{r \in \Z/87\Z} \chi_{87}(8,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right) = 0.0 \)