Properties

Conductor 95
Order 18
Real No
Primitive Yes
Parity Odd
Orbit Label 95.o

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(95)
 
sage: chi = H[59]
 
pari: [g,chi] = znchar(Mod(59,95))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 95
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 18
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 = Odd
Orbit label = 95.o
Orbit index = 15

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 ]
 

\(\chi_{95}(14,\cdot)\) \(\chi_{95}(29,\cdot)\) \(\chi_{95}(34,\cdot)\) \(\chi_{95}(59,\cdot)\) \(\chi_{95}(79,\cdot)\) \(\chi_{95}(89,\cdot)\)

Values on generators

\((77,21)\) → \((-1,e\left(\frac{1}{18}\right))\)

Values

-112346789111213
\(-1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{9}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 95 }(59,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{95}(59,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(59,r) e\left(\frac{2r}{95}\right) = -9.2337567792+-3.1205345282i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 95 }(59,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{95}(59,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(59,r) \chi_{95}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 95 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{95}(59,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(59,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = 0.0 \)