Properties

Label 1.1
Modulus $1$
Conductor $1$
Order $1$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath

Basic properties

Modulus: \(1\)
Conductor: \(1\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
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 1.a

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

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

Values

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

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q\)

Gauss sum

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

Jacobi sum

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

Kloosterman sum

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