Properties

Conductor 287
Order 15
Real no
Primitive no
Minimal yes
Parity even
Orbit label 861.bk

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(861)
 
sage: chi = H[16]
 
pari: [g,chi] = znchar(Mod(16,861))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 15
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 = 861.bk
Orbit index = 37

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 ]
 

\(\chi_{861}(16,\cdot)\) \(\chi_{861}(37,\cdot)\) \(\chi_{861}(100,\cdot)\) \(\chi_{861}(256,\cdot)\) \(\chi_{861}(508,\cdot)\) \(\chi_{861}(529,\cdot)\) \(\chi_{861}(592,\cdot)\) \(\chi_{861}(625,\cdot)\)

Values on generators

\((575,493,211)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right))\)

Values

-112458101113161719
\(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{15}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{861}(16,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(16,r) e\left(\frac{2r}{861}\right) = 12.1828162597+-11.7719576954i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 861 }(16,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{861}(16,\cdot),\chi_{861}(1,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(16,r) \chi_{861}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{861}(16,·)) = \sum_{r \in \Z/861\Z} \chi_{861}(16,r) e\left(\frac{1 r + 2 r^{-1}}{861}\right) = -5.2477389287+-5.8282045287i \)