Properties

Modulus 35
Conductor 1
Order 1
Real yes
Primitive no
Minimal yes
Parity even
Orbit label 35.a

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(35)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0]))
 
pari: [g,chi] = znchar(Mod(1,35))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 35
Conductor = 1
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 1
Real = yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 35.a
Orbit index = 1

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{35}(1,\cdot)\)

Values on generators

\((22,31)\) → \((1,1)\)

Values

-1123468911121316
\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 35 }(1,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(1,r) e\left(\frac{2r}{35}\right) = 1.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 35 }(1,·),\chi_{ 35 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{35}(1,\cdot),\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(1,r) \chi_{35}(1,1-r) = 15 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 35 }(1,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{35}(1,·)) = \sum_{r \in \Z/35\Z} \chi_{35}(1,r) e\left(\frac{1 r + 2 r^{-1}}{35}\right) = -3.3275215358 \)