Properties

Label 71.17
Modulus $71$
Conductor $71$
Order $10$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(71)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(17,71))
 

Basic properties

Modulus: \(71\)
Conductor: \(71\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 71.e

\(\chi_{71}(14,\cdot)\) \(\chi_{71}(17,\cdot)\) \(\chi_{71}(46,\cdot)\) \(\chi_{71}(66,\cdot)\)

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

Values on generators

\(7\) → \(e\left(\frac{7}{10}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: 10.0.45848500718449031.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 71 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{71}(17,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(17,r) e\left(\frac{2r}{71}\right) = 0.6851212954+-8.3982503422i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 71 }(17,·),\chi_{ 71 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{71}(17,\cdot),\chi_{71}(1,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(17,r) \chi_{71}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 71 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{71}(17,·)) = \sum_{r \in \Z/71\Z} \chi_{71}(17,r) e\left(\frac{1 r + 2 r^{-1}}{71}\right) = 0.4956423809+-0.6821932121i \)