Properties

Label 469.330
Modulus $469$
Conductor $67$
Order $11$
Real no
Primitive no
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(469, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,16]))
 
pari: [g,chi] = znchar(Mod(330,469))
 

Basic properties

Modulus: \(469\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{67}(62,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 469.u

\(\chi_{469}(15,\cdot)\) \(\chi_{469}(22,\cdot)\) \(\chi_{469}(64,\cdot)\) \(\chi_{469}(92,\cdot)\) \(\chi_{469}(148,\cdot)\) \(\chi_{469}(225,\cdot)\) \(\chi_{469}(260,\cdot)\) \(\chi_{469}(330,\cdot)\) \(\chi_{469}(344,\cdot)\) \(\chi_{469}(442,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 11.11.1822837804551761449.1

Values on generators

\((269,337)\) → \((1,e\left(\frac{8}{11}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 469 }(330,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{469}(330,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(330,r) e\left(\frac{2r}{469}\right) = -0.5557608196+8.1664637335i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 469 }(330,·),\chi_{ 469 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{469}(330,\cdot),\chi_{469}(1,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(330,r) \chi_{469}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 469 }(330,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{469}(330,·)) = \sum_{r \in \Z/469\Z} \chi_{469}(330,r) e\left(\frac{1 r + 2 r^{-1}}{469}\right) = 3.0519159947+-3.5220988134i \)