Properties

Label 32.13
Modulus $32$
Conductor $32$
Order $8$
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(32, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,7]))
 
pari: [g,chi] = znchar(Mod(13,32))
 

Basic properties

Modulus: \(32\)
Conductor: \(32\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 32.g

\(\chi_{32}(5,\cdot)\) \(\chi_{32}(13,\cdot)\) \(\chi_{32}(21,\cdot)\) \(\chi_{32}(29,\cdot)\)

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

\((31,5)\) → \((1,e\left(\frac{7}{8}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{32})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 32 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{32}(13,\cdot)) = \sum_{r\in \Z/32\Z} \chi_{32}(13,r) e\left(\frac{r}{16}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 32 }(13,·),\chi_{ 32 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{32}(13,\cdot),\chi_{32}(1,\cdot)) = \sum_{r\in \Z/32\Z} \chi_{32}(13,r) \chi_{32}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 32 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{32}(13,·)) = \sum_{r \in \Z/32\Z} \chi_{32}(13,r) e\left(\frac{1 r + 2 r^{-1}}{32}\right) = 5.5481593813+-1.1035975171i \)