Properties

Conductor 17
Order 4
Real no
Primitive no
Minimal yes
Parity even
Orbit label 51.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(51)
 
sage: chi = H[4]
 
pari: [g,chi] = znchar(Mod(4,51))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 17
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 51.e
Orbit index = 5

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 ]
 

\(\chi_{51}(4,\cdot)\) \(\chi_{51}(13,\cdot)\)

Values on generators

\((35,37)\) → \((1,-i)\)

Values

-11245781011131416
\(1\)\(1\)\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(i\)\(i\)\(1\)\(-i\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 51 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{51}(4,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(4,r) e\left(\frac{2r}{51}\right) = -3.2498542756+-2.5374095427i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 51 }(4,·),\chi_{ 51 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{51}(4,\cdot),\chi_{51}(1,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(4,r) \chi_{51}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 51 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{51}(4,·)) = \sum_{r \in \Z/51\Z} \chi_{51}(4,r) e\left(\frac{1 r + 2 r^{-1}}{51}\right) = 6.5583773584i \)