Properties

Modulus 60
Conductor 3
Order 2
Real yes
Primitive no
Minimal yes
Parity odd
Orbit label 60.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(60)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,0]))
 
pari: [g,chi] = znchar(Mod(41,60))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 60
Conductor = 3
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2
Real = yes
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 = odd
Orbit label = 60.g
Orbit index = 7

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 ]
 

\(\chi_{60}(41,\cdot)\)

Values on generators

\((31,41,37)\) → \((1,-1,1)\)

Values

-117111317192329313741
\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 60 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{60}(41,\cdot)) = \sum_{r\in \Z/60\Z} \chi_{60}(41,r) e\left(\frac{r}{30}\right) = 3.4641016151i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 60 }(41,·),\chi_{ 60 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{60}(41,\cdot),\chi_{60}(1,\cdot)) = \sum_{r\in \Z/60\Z} \chi_{60}(41,r) \chi_{60}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 60 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{60}(41,·)) = \sum_{r \in \Z/60\Z} \chi_{60}(41,r) e\left(\frac{1 r + 2 r^{-1}}{60}\right) = 0.0 \)