Properties

Label 216.67
Modulus $216$
Conductor $216$
Order $18$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,9,8]))
 
pari: [g,chi] = znchar(Mod(67,216))
 

Basic properties

Modulus: \(216\)
Conductor: \(216\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 216.r

\(\chi_{216}(43,\cdot)\) \(\chi_{216}(67,\cdot)\) \(\chi_{216}(115,\cdot)\) \(\chi_{216}(139,\cdot)\) \(\chi_{216}(187,\cdot)\) \(\chi_{216}(211,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 18.0.132173713091594538512566714368.2

Values on generators

\((55,109,137)\) → \((-1,-1,e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{7}{18}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 216 }(67,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{216}(67,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(67,r) e\left(\frac{r}{108}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 216 }(67,·),\chi_{ 216 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{216}(67,\cdot),\chi_{216}(1,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(67,r) \chi_{216}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 216 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{216}(67,·)) = \sum_{r \in \Z/216\Z} \chi_{216}(67,r) e\left(\frac{1 r + 2 r^{-1}}{216}\right) = 0.0 \)