Properties

Label 35.19
Modulus $35$
Conductor $35$
Order $6$
Real no
Primitive yes
Minimal yes
Parity odd

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([3,5]))
 
pari: [g,chi] = znchar(Mod(19,35))
 

Basic properties

Modulus: \(35\)
Conductor: \(35\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 35.i

\(\chi_{35}(19,\cdot)\) \(\chi_{35}(24,\cdot)\)

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

\((22,31)\) → \((-1,e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.2100875.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 35 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{35}(19,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(19,r) e\left(\frac{2r}{35}\right) = -5.4563040634+-2.2866451337i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 35 }(19,·),\chi_{ 35 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{35}(19,\cdot),\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(19,r) \chi_{35}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 35 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{35}(19,·)) = \sum_{r \in \Z/35\Z} \chi_{35}(19,r) e\left(\frac{1 r + 2 r^{-1}}{35}\right) = 0.0 \)