Properties

Label 16.7
Modulus $16$
Conductor $8$
Order $2$
Real yes
Primitive no
Minimal no
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(16)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(7,16))
 

Basic properties

Modulus: \(16\)
Conductor: \(8\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{8}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 16.d

\(\chi_{16}(7,\cdot)\)

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

\((15,5)\) → \((-1,-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)
\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-2}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 16 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{16}(7,\cdot)) = \sum_{r\in \Z/16\Z} \chi_{16}(7,r) e\left(\frac{r}{8}\right) = 5.6568542495i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 16 }(7,·),\chi_{ 16 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{16}(7,\cdot),\chi_{16}(1,\cdot)) = \sum_{r\in \Z/16\Z} \chi_{16}(7,r) \chi_{16}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 16 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{16}(7,·)) = \sum_{r \in \Z/16\Z} \chi_{16}(7,r) e\left(\frac{1 r + 2 r^{-1}}{16}\right) = -0.0 \)