Properties

Label 7.6
Modulus $7$
Conductor $7$
Order $2$
Real yes
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(7)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(6,7))
 

Kronecker symbol representation

sage: kronecker_character(-7)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-7}{\bullet}\right)\)

Basic properties

Modulus: \(7\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
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 7.b

\(\chi_{7}(6,\cdot)\)

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

\(3\) → \(-1\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)
\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-7}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 7 }(6,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{7}(6,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) e\left(\frac{2r}{7}\right) = 2.6457513111i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 7 }(6,·),\chi_{ 7 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{7}(6,\cdot),\chi_{7}(1,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) \chi_{7}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 7 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{7}(6,·)) = \sum_{r \in \Z/7\Z} \chi_{7}(6,r) e\left(\frac{1 r + 2 r^{-1}}{7}\right) = 3.2991979214i \)