Properties

Modulus 51
Conductor 51
Order 16
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 51.i

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(51)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,11]))
 
pari: [g,chi] = znchar(Mod(41,51))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 51
Conductor = 51
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 16
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 = 51.i
Orbit index = 9

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 ]
 

\(\chi_{51}(5,\cdot)\) \(\chi_{51}(11,\cdot)\) \(\chi_{51}(14,\cdot)\) \(\chi_{51}(20,\cdot)\) \(\chi_{51}(23,\cdot)\) \(\chi_{51}(29,\cdot)\) \(\chi_{51}(41,\cdot)\) \(\chi_{51}(44,\cdot)\)

Values on generators

\((35,37)\) → \((-1,e\left(\frac{11}{16}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(-i\)\(e\left(\frac{11}{16}\right)\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{16})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 51 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{51}(41,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(41,r) e\left(\frac{2r}{51}\right) = 7.0449771312+1.1697423734i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 51 }(41,·),\chi_{ 51 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{51}(41,\cdot),\chi_{51}(1,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(41,r) \chi_{51}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 51 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{51}(41,·)) = \sum_{r \in \Z/51\Z} \chi_{51}(41,r) e\left(\frac{1 r + 2 r^{-1}}{51}\right) = -0.0 \)