Properties

Modulus 85
Conductor 85
Order 8
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 85.n

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(85)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,3]))
 
pari: [g,chi] = znchar(Mod(83,85))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 85
Conductor = 85
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 85.n
Orbit index = 14

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{85}(42,\cdot)\) \(\chi_{85}(53,\cdot)\) \(\chi_{85}(77,\cdot)\) \(\chi_{85}(83,\cdot)\)

Values on generators

\((52,71)\) → \((-i,e\left(\frac{3}{8}\right))\)

Values

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

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 85 }(83,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{85}(83,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(83,r) e\left(\frac{2r}{85}\right) = 9.1077819982+1.4311907888i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 85 }(83,·),\chi_{ 85 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{85}(83,\cdot),\chi_{85}(1,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(83,r) \chi_{85}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 85 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{85}(83,·)) = \sum_{r \in \Z/85\Z} \chi_{85}(83,r) e\left(\frac{1 r + 2 r^{-1}}{85}\right) = -5.650434781i \)