Properties

Label 189.31
Modulus $189$
Conductor $189$
Order $18$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(189, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3]))
 
pari: [g,chi] = znchar(Mod(31,189))
 

Basic properties

Modulus: \(189\)
Conductor: \(189\)
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 189.z

\(\chi_{189}(31,\cdot)\) \(\chi_{189}(61,\cdot)\) \(\chi_{189}(94,\cdot)\) \(\chi_{189}(124,\cdot)\) \(\chi_{189}(157,\cdot)\) \(\chi_{189}(187,\cdot)\)

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: 18.0.4675260431318755700605016170035783.2

Values on generators

\((29,136)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 189 }(31,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{189}(31,\cdot)) = \sum_{r\in \Z/189\Z} \chi_{189}(31,r) e\left(\frac{2r}{189}\right) = -10.6226595866+8.7269183168i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 189 }(31,·),\chi_{ 189 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{189}(31,\cdot),\chi_{189}(1,\cdot)) = \sum_{r\in \Z/189\Z} \chi_{189}(31,r) \chi_{189}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 189 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{189}(31,·)) = \sum_{r \in \Z/189\Z} \chi_{189}(31,r) e\left(\frac{1 r + 2 r^{-1}}{189}\right) = 6.1679094353+-1.087568848i \)