Properties

Label 71.48
Modulus $71$
Conductor $71$
Order $7$
Real no
Primitive yes
Minimal yes
Parity even

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([5]))
 
pari: [g,chi] = znchar(Mod(48,71))
 

Basic properties

Modulus: \(71\)
Conductor: \(71\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 71.d

\(\chi_{71}(20,\cdot)\) \(\chi_{71}(30,\cdot)\) \(\chi_{71}(32,\cdot)\) \(\chi_{71}(37,\cdot)\) \(\chi_{71}(45,\cdot)\) \(\chi_{71}(48,\cdot)\)

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

\(7\) → \(e\left(\frac{5}{7}\right)\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 7.7.128100283921.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 71 }(48,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{71}(48,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(48,r) e\left(\frac{2r}{71}\right) = 0.7491649495+-8.3927797468i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 71 }(48,·),\chi_{ 71 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{71}(48,\cdot),\chi_{71}(1,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(48,r) \chi_{71}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 71 }(48,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{71}(48,·)) = \sum_{r \in \Z/71\Z} \chi_{71}(48,r) e\left(\frac{1 r + 2 r^{-1}}{71}\right) = -0.6141077175+-0.7700667208i \)