Properties

Label 690.689
Modulus $690$
Conductor $345$
Order $2$
Real yes
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(690, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1,1]))
 
pari: [g,chi] = znchar(Mod(689,690))
 

Basic properties

Modulus: \(690\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{345}(344,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 690.h

\(\chi_{690}(689,\cdot)\)

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

\((461,277,511)\) → \((-1,-1,-1)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{345}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 690 }(689,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{690}(689,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(689,r) e\left(\frac{r}{345}\right) = 18.574175621 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 690 }(689,·),\chi_{ 690 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{690}(689,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(689,r) \chi_{690}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 690 }(689,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{690}(689,·)) = \sum_{r \in \Z/690\Z} \chi_{690}(689,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right) = 0.0 \)