Properties

Label 216.71
Modulus $216$
Conductor $36$
Order $6$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(216)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,0,5]))
 
pari: [g,chi] = znchar(Mod(71,216))
 

Basic properties

Modulus: \(216\)
Conductor: \(36\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{36}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 216.o

\(\chi_{216}(71,\cdot)\) \(\chi_{216}(143,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((55,109,137)\) → \((-1,1,e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{36})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 216 }(71,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{216}(71,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(71,r) e\left(\frac{r}{108}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 216 }(71,·),\chi_{ 216 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{216}(71,\cdot),\chi_{216}(1,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(71,r) \chi_{216}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 216 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{216}(71,·)) = \sum_{r \in \Z/216\Z} \chi_{216}(71,r) e\left(\frac{1 r + 2 r^{-1}}{216}\right) = -0.0 \)