Properties

Modulus 76
Conductor 19
Order 3
Real no
Primitive no
Minimal yes
Parity even
Orbit label 76.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(76)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(45,76))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 76
Conductor = 19
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 3
Real = no
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 = 76.e
Orbit index = 5

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 ]
 

\(\chi_{76}(45,\cdot)\) \(\chi_{76}(49,\cdot)\)

Values on generators

\((39,21)\) → \((1,e\left(\frac{1}{3}\right))\)

Values

-113579111315172123
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 76 }(45,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{76}(45,\cdot)) = \sum_{r\in \Z/76\Z} \chi_{76}(45,r) e\left(\frac{r}{38}\right) = 5.8554274455+6.4586352607i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 76 }(45,·),\chi_{ 76 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{76}(45,\cdot),\chi_{76}(1,\cdot)) = \sum_{r\in \Z/76\Z} \chi_{76}(45,r) \chi_{76}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 76 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{76}(45,·)) = \sum_{r \in \Z/76\Z} \chi_{76}(45,r) e\left(\frac{1 r + 2 r^{-1}}{76}\right) = -0.0 \)