Properties

Label 65.29
Modulus $65$
Conductor $65$
Order $6$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(65, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,2]))
 
pari: [g,chi] = znchar(Mod(29,65))
 

Basic properties

Modulus: \(65\)
Conductor: \(65\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
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 65.n

\(\chi_{65}(9,\cdot)\) \(\chi_{65}(29,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.3570125.1

Values on generators

\((27,41)\) → \((-1,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 65 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{65}(29,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(29,r) e\left(\frac{2r}{65}\right) = 5.7373127892+-5.6642070901i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 65 }(29,·),\chi_{ 65 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{65}(29,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(29,r) \chi_{65}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 65 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{65}(29,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(29,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 0.0 \)