Properties

Label 133.9
Modulus $133$
Conductor $133$
Order $9$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(133, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,8]))
 
pari: [g,chi] = znchar(Mod(9,133))
 

Basic properties

Modulus: \(133\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 133.u

\(\chi_{133}(9,\cdot)\) \(\chi_{133}(23,\cdot)\) \(\chi_{133}(44,\cdot)\) \(\chi_{133}(74,\cdot)\) \(\chi_{133}(81,\cdot)\) \(\chi_{133}(130,\cdot)\)

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

\((115,78)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.1998099208210609.2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 133 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{133}(9,\cdot)) = \sum_{r\in \Z/133\Z} \chi_{133}(9,r) e\left(\frac{2r}{133}\right) = 9.5000977597+6.5382063715i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 133 }(9,·),\chi_{ 133 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{133}(9,\cdot),\chi_{133}(1,\cdot)) = \sum_{r\in \Z/133\Z} \chi_{133}(9,r) \chi_{133}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 133 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{133}(9,·)) = \sum_{r \in \Z/133\Z} \chi_{133}(9,r) e\left(\frac{1 r + 2 r^{-1}}{133}\right) = -4.5656891402+-1.661774946i \)