Properties

Conductor 32
Order 8
Real No
Primitive No
Parity Even
Orbit Label 640.x

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(640)
 
sage: chi = H[561]
 
pari: [g,chi] = znchar(Mod(561,640))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 32
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 640.x
Orbit index = 24

Galois orbit

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

\(\chi_{640}(81,\cdot)\) \(\chi_{640}(241,\cdot)\) \(\chi_{640}(401,\cdot)\) \(\chi_{640}(561,\cdot)\)

Inducing primitive character

\(\chi_{32}(21,\cdot)\)

Values on generators

\((511,261,257)\) → \((1,e\left(\frac{5}{8}\right),1)\)

Values

-1137911131719212327
\(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 640 }(561,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{640}(561,\cdot)) = \sum_{r\in \Z/640\Z} \chi_{640}(561,r) e\left(\frac{r}{320}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 640 }(561,·),\chi_{ 640 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{640}(561,\cdot),\chi_{640}(1,\cdot)) = \sum_{r\in \Z/640\Z} \chi_{640}(561,r) \chi_{640}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 640 }(561,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{640}(561,·)) = \sum_{r \in \Z/640\Z} \chi_{640}(561,r) e\left(\frac{1 r + 2 r^{-1}}{640}\right) = 0.0 \)