Properties

Label 80.69
Modulus $80$
Conductor $80$
Order $4$
Real no
Primitive yes
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(80)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,2]))
 
pari: [g,chi] = znchar(Mod(69,80))
 

Basic properties

Modulus: \(80\)
Conductor: \(80\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
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 80.q

\(\chi_{80}(29,\cdot)\) \(\chi_{80}(69,\cdot)\)

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

\((31,21,17)\) → \((1,i,-1)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(i\)\(1\)\(-1\)\(i\)\(i\)\(-1\)\(-i\)\(i\)\(1\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.51200.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 80 }(69,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{80}(69,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(69,r) e\left(\frac{r}{40}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 80 }(69,·),\chi_{ 80 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{80}(69,\cdot),\chi_{80}(1,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(69,r) \chi_{80}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 80 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{80}(69,·)) = \sum_{r \in \Z/80\Z} \chi_{80}(69,r) e\left(\frac{1 r + 2 r^{-1}}{80}\right) = -0.0 \)