Properties

Conductor 861
Order 60
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 861.ch

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[2]
 
pari: [g,chi] = znchar(Mod(2,861))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 861
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
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 = odd
Orbit label = 861.ch
Orbit index = 60

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 ]
 

\(\chi_{861}(2,\cdot)\) \(\chi_{861}(74,\cdot)\) \(\chi_{861}(128,\cdot)\) \(\chi_{861}(200,\cdot)\) \(\chi_{861}(254,\cdot)\) \(\chi_{861}(326,\cdot)\) \(\chi_{861}(389,\cdot)\) \(\chi_{861}(431,\cdot)\) \(\chi_{861}(443,\cdot)\) \(\chi_{861}(494,\cdot)\) \(\chi_{861}(569,\cdot)\) \(\chi_{861}(620,\cdot)\) \(\chi_{861}(695,\cdot)\) \(\chi_{861}(746,\cdot)\) \(\chi_{861}(758,\cdot)\) \(\chi_{861}(800,\cdot)\)

Values on generators

\((575,493,211)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right))\)

Values

-112458101113161719
\(-1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{31}{60}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{861}(2,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(2,r) e\left(\frac{2r}{861}\right) = 15.9538713094+-24.6266926371i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 861 }(2,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{861}(2,\cdot),\chi_{861}(1,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(2,r) \chi_{861}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{861}(2,·)) = \sum_{r \in \Z/861\Z} \chi_{861}(2,r) e\left(\frac{1 r + 2 r^{-1}}{861}\right) = 0.0 \)