Properties

Modulus 128
Conductor 128
Order 32
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 128.k

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(128)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(5,128))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 128
Conductor = 128
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 32
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 128.k
Orbit index = 11

Galois orbit

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

\(\chi_{128}(5,\cdot)\) \(\chi_{128}(13,\cdot)\) \(\chi_{128}(21,\cdot)\) \(\chi_{128}(29,\cdot)\) \(\chi_{128}(37,\cdot)\) \(\chi_{128}(45,\cdot)\) \(\chi_{128}(53,\cdot)\) \(\chi_{128}(61,\cdot)\) \(\chi_{128}(69,\cdot)\) \(\chi_{128}(77,\cdot)\) \(\chi_{128}(85,\cdot)\) \(\chi_{128}(93,\cdot)\) \(\chi_{128}(101,\cdot)\) \(\chi_{128}(109,\cdot)\) \(\chi_{128}(117,\cdot)\) \(\chi_{128}(125,\cdot)\)

Values on generators

\((127,5)\) → \((1,e\left(\frac{1}{32}\right))\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{13}{32}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{32})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 128 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{128}(5,\cdot)) = \sum_{r\in \Z/128\Z} \chi_{128}(5,r) e\left(\frac{r}{64}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 128 }(5,·),\chi_{ 128 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{128}(5,\cdot),\chi_{128}(1,\cdot)) = \sum_{r\in \Z/128\Z} \chi_{128}(5,r) \chi_{128}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 128 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{128}(5,·)) = \sum_{r \in \Z/128\Z} \chi_{128}(5,r) e\left(\frac{1 r + 2 r^{-1}}{128}\right) = 9.0872558754+6.7395682841i \)