Properties

Modulus 73
Conductor 1
Order 1
Real yes
Primitive no
Minimal yes
Parity even
Orbit label 73.a

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(73)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0]))
 
pari: [g,chi] = znchar(Mod(1,73))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 73
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 = 73.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 ]
 

\(\chi_{73}(1,\cdot)\)

Values on generators

\(5\) → \(1\)

Values

-11234567891011
\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 73 }(1,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{73}(1,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(1,r) e\left(\frac{2r}{73}\right) = -1.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 73 }(1,·),\chi_{ 73 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{73}(1,\cdot),\chi_{73}(1,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(1,r) \chi_{73}(1,1-r) = 71 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 73 }(1,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{73}(1,·)) = \sum_{r \in \Z/73\Z} \chi_{73}(1,r) e\left(\frac{1 r + 2 r^{-1}}{73}\right) = -2.2629726532 \)